GCSE Maths: Multi-stage ratio problems with algebra (Higher)
Part of the GCSE Maths ratio and proportion chapter (Higher). Chapter overview · whole chapter on one page · Download worksheet (PDF) · Answers (PDF)
Multi-stage ratio problems with algebra
In harder problems the numbers change: items are added or removed, money is given away, or people get older. The original numbers are in one ratio and the new numbers are in another. We cannot just divide the total, because we do not know it. Instead we use algebra.
The method. (1) If the original ratio is 5 : 7, write the original amounts as 5x and 7x, where x is the value of one part. This keeps the original ratio exactly right for every value of x. (2) Write expressions for the amounts after the change: for example 6 more of the first gives 5x + 6, and 7x stays the same. (3) Use the new ratio, say 6 : 7, to form an equation: 5x + 67x = 67. (4) Cross-multiply: 7(5x + 6) = 6 × 7x, expand and solve for x. (5) Put x back to find the amounts the question asks for, and check that the new amounts really are in the new ratio.
Transfers. If money is passed from one person to another, subtract from one and add to the other: 5x − 4 and 3x + 4. The total does not change. Ages. In 6 years’ time everyone is 6 years older, so add 6 to every age. Expressions in a ratio. If x : y = 3 : 5, write x = 3k and y = 5k. Then any expression in x and y becomes an expression in k, and k cancels from the ratio. Equations from ratios. A ratio with letters in it, such as (3x + 1) : (2x − 1) = 5 : 3, is solved by cross-multiplying.
Key ideas
- Original ratio p : q means the amounts are px and qx.
- Write expressions for the new amounts, then form an equation from the new ratio by cross-multiplying.
- Transfers: subtract from one amount, add to the other. The total stays the same.
- Ages: add the same number of years to every age.
- If x : y = p : q, write x = pk and y = qk. Check the answer by testing the new ratio.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 5 : 7 means the boys is 5 equal parts and the girls is 7 equal parts. Let one part be x.Boys = 5x
Girls = 7x - Write expressions for the amounts after the change. Six boys join, so add 6 to the number of boys. The number of girls does not change.Boys = 5x + 6
Girls = 7x - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.5x + 67x = 67
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 7 and by 7x.) Write each product in brackets.7 × (5x + 6) = 6 × 7x
- Expand the brackets. Multiply every term inside each bracket by the number outside.35x + 42 = 42x
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.35x + 42 = 42x
Subtract 35x from both sides: 42 = 7x - Divide both sides by 7. 7x means 7 multiplied by x, so the inverse is dividing by 7.x = 42 ÷ 7 = 6
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original boys: 5 × 6 = 30
Original girls: 7 × 6 = 42
Check. After the change: boys = 36 and girls = 42. 36 : 42 simplifies (÷ 6) to 6 : 7, which is the ratio 6 : 7 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 5 : 3 means the Sam is 5 equal parts and the Tia is 3 equal parts. Let one part be x.Sam = 5x
Tia = 3x - Write expressions for the amounts after the gift. Sam gives away £4, so subtract 4 from Sam's amount. Tia receives £4, so add 4 to Tia's amount.Sam = 5x − 4
Tia = 3x + 4 - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.5x − 43x + 4 = 1311
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 11 and by (3x + 4).) Write each product in brackets.11 × (5x − 4) = 13 × (3x + 4)
- Expand the brackets. Multiply every term inside each bracket by the number outside.55x − 44 = 39x + 52
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.55x − 44 = 39x + 52
Subtract 39x from both sides: 16x − 44 = 52
Add 44 to both sides: 16x = 96 - Divide both sides by 16. 16x means 16 multiplied by x, so the inverse is dividing by 16.x = 96 ÷ 16 = 6
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original Sam: 5 × 6 = £30
Original Tia: 3 × 6 = £18 - Add to find the total. The question asks for the total.£30 + £18 = £48
Check. After the gift: Sam = £26 and Tia = £22. £26 : £22 simplifies (÷ 2) to 13 : 11, which is the ratio 13 : 11 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. Cross-multiplying turns a = b over c = d into a × d = b × c.
Method. Write the ratio as a fraction, cross-multiply to remove the fractions, expand and solve the linear equation.
- Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.3x + 12x − 1 = 53
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 3 and by (2x − 1).) Write each product in brackets.3 × (3x + 1) = 5 × (2x − 1)
- Expand the brackets. Multiply every term inside each bracket by the number outside.9x + 3 = 10x − 5
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.9x + 3 = 10x − 5
Subtract 9x from both sides: 3 = x − 5
Add 5 to both sides: 8 = x - Divide both sides by 1. 1x means 1 multiplied by x, so the inverse is dividing by 1.x = 8 ÷ 1 = 8
Check. Substitute x = 8: 3x + 1 = 25 and 2x − 1 = 15. 25 : 15 = 5 : 3 ✓
Watch out. Cross-multiply carefully: top of one fraction times bottom of the other. Put the expressions in brackets before you multiply.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Write x and y as multiples of one letter k, substitute, then cancel k.
- Write x and y using one letter. The ratio 3 : 5 means x is 3 parts and y is 5 parts. If one part is k, then x = 3k and y = 5k.x = 3k
y = 5k - Substitute into the first expression. Replace x and y by their values in k.2x + y = 2(3k) + (1)(5k) = 6k + (5)k = 11k
- Substitute into the second expression. Do the same for the second expression.x + 3y = 1(3k) + (3)(5k) = 3k + (15)k = 18k
- Write the ratio and cancel k. Both terms contain k, so k divides out. The ratio of the numbers is what remains.11k : 18k = 11 : 18
Check. Choose numbers in the ratio, e.g. x = 9, y = 15: 2x + y = 33 and x + 3y = 54; the ratio 33 : 54 simplifies to 11 : 18. ✓
Watch out. Cancel k only at the end, when it is a factor of both parts.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 3 : 5 means the Ali is 3 equal parts and the Bea is 5 equal parts. Let one part be x.Ali = 3x
Bea = 5x - Write expressions for the amounts in 6 years. In 6 years each person is 6 years older, so add 6 to both ages.Ali = 3x + 6
Bea = 5x + 6 - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.3x + 65x + 6 = 57
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 7 and by (5x + 6).) Write each product in brackets.7 × (3x + 6) = 5 × (5x + 6)
- Expand the brackets. Multiply every term inside each bracket by the number outside.21x + 42 = 25x + 30
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.21x + 42 = 25x + 30
Subtract 21x from both sides: 42 = 4x + 30
Subtract 30 from both sides: 12 = 4x - Divide both sides by 4. 4x means 4 multiplied by x, so the inverse is dividing by 4.x = 12 ÷ 4 = 3
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original Ali: 3 × 3 = 9 years
Original Bea: 5 × 3 = 15 years
Check. In 6 years: Ali = 15 years and Bea = 21 years. 15 years : 21 years simplifies (÷ 3) to 5 : 7, which is the ratio 5 : 7 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Watch out
The new ratio only applies to the amounts after the change. Do not write the new ratio as 5x : 7x with the new numbers; use the new expressions such as (5x + 6) and 7x. And always check that the answer fits the new ratio.Exercise 15I Multi-stage ratio problems
Non-calculatorKey words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 3 : 4 means the red counters is 3 equal parts and the blue counters is 4 equal parts. Let one part be x.Red counters = 3x
Blue counters = 4x - Write expressions for the amounts after the change. Six red counters are added, so add 6 to the red counters.Red counters = 3x + 6
Blue counters = 4x - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.3x + 64x = 32
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 2 and by 4x.) Write each product in brackets.2 × (3x + 6) = 3 × 4x
- Expand the brackets. Multiply every term inside each bracket by the number outside.6x + 12 = 12x
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.6x + 12 = 12x
Subtract 6x from both sides: 12 = 6x - Divide both sides by 6. 6x means 6 multiplied by x, so the inverse is dividing by 6.x = 12 ÷ 6 = 2
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original red counters: 3 × 2 = 6
Original blue counters: 4 × 2 = 8
Check. After the change: red counters = 12 and blue counters = 8. 12 : 8 simplifies (÷ 4) to 3 : 2, which is the ratio 3 : 2 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 5 : 7 means the boys is 5 equal parts and the girls is 7 equal parts. Let one part be x.Boys = 5x
Girls = 7x - Write expressions for the amounts after the change. Six boys join, so add 6 to the boys.Boys = 5x + 6
Girls = 7x - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.5x + 67x = 67
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 7 and by 7x.) Write each product in brackets.7 × (5x + 6) = 6 × 7x
- Expand the brackets. Multiply every term inside each bracket by the number outside.35x + 42 = 42x
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.35x + 42 = 42x
Subtract 35x from both sides: 42 = 7x - Divide both sides by 7. 7x means 7 multiplied by x, so the inverse is dividing by 7.x = 42 ÷ 7 = 6
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original boys: 5 × 6 = 30
Original girls: 7 × 6 = 42
Check. After the change: boys = 36 and girls = 42. 36 : 42 simplifies (÷ 6) to 6 : 7, which is the ratio 6 : 7 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 7 : 4 means the cats is 7 equal parts and the dogs is 4 equal parts. Let one part be x.Cats = 7x
Dogs = 4x - Write expressions for the amounts after the change. Nine cats leave, so subtract 9 from the cats.Cats = 7x − 9
Dogs = 4x - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.7x − 94x = 118
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 8 and by 4x.) Write each product in brackets.8 × (7x − 9) = 11 × 4x
- Expand the brackets. Multiply every term inside each bracket by the number outside.56x − 72 = 44x
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.56x − 72 = 44x
Subtract 44x from both sides: 12x − 72 = 0
Add 72 to both sides: 12x = 72 - Divide both sides by 12. 12x means 12 multiplied by x, so the inverse is dividing by 12.x = 72 ÷ 12 = 6
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original cats: 7 × 6 = 42
Original dogs: 4 × 6 = 24
Check. After the change: cats = 33 and dogs = 24. 33 : 24 simplifies (÷ 3) to 11 : 8, which is the ratio 11 : 8 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 2 : 5 means the A is 2 equal parts and the B is 5 equal parts. Let one part be x.A = 2x
B = 5x - Write expressions for the amounts after the change. 10 is added to each number.A = 2x + 10
B = 5x + 10 - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.2x + 105x + 10 = 12
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 2 and by (5x + 10).) Write each product in brackets.2 × (2x + 10) = 1 × (5x + 10)
- Expand the brackets. Multiply every term inside each bracket by the number outside.4x + 20 = 5x + 10
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.4x + 20 = 5x + 10
Subtract 4x from both sides: 20 = x + 10
Subtract 10 from both sides: 10 = x - Divide both sides by 1. 1x means 1 multiplied by x, so the inverse is dividing by 1.x = 10 ÷ 1 = 10
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original A: 2 × 10 = 20
Original B: 5 × 10 = 50
Check. After the change: A = 30 and B = 60. 30 : 60 simplifies (÷ 30) to 1 : 2, which is the ratio 1 : 2 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 7 : 3 means the Ali is 7 equal parts and the Ben is 3 equal parts. Let one part be x.Ali = 7x
Ben = 3x - Write expressions for the amounts after the gift. Ali gives 8 marbles to Ben, so subtract 8 from Ali and add 8 to Ben.Ali = 7x − 8
Ben = 3x + 8 - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.7x − 83x + 8 = 11
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 1 and by (3x + 8).) Write each product in brackets.1 × (7x − 8) = 1 × (3x + 8)
- Expand the brackets. Multiply every term inside each bracket by the number outside.7x − 8 = 3x + 8
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.7x − 8 = 3x + 8
Subtract 3x from both sides: 4x − 8 = 8
Add 8 to both sides: 4x = 16 - Divide both sides by 4. 4x means 4 multiplied by x, so the inverse is dividing by 4.x = 16 ÷ 4 = 4
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original Ali: 7 × 4 = 28
Original Ben: 3 × 4 = 12
Check. After the gift: Ali = 20 and Ben = 20. 20 : 20 simplifies (÷ 20) to 1 : 1, which is the ratio 1 : 1 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 4 : 5 means the Ana is 4 equal parts and the Bo is 5 equal parts. Let one part be x.Ana = 4x
Bo = 5x - Write expressions for the amounts after the gift. Ana gives £3 to Bo, so subtract 3 from Ana and add 3 to Bo.Ana = 4x − 3
Bo = 5x + 3 - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.4x − 35x + 3 = 711
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 11 and by (5x + 3).) Write each product in brackets.11 × (4x − 3) = 7 × (5x + 3)
- Expand the brackets. Multiply every term inside each bracket by the number outside.44x − 33 = 35x + 21
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.44x − 33 = 35x + 21
Subtract 35x from both sides: 9x − 33 = 21
Add 33 to both sides: 9x = 54 - Divide both sides by 9. 9x means 9 multiplied by x, so the inverse is dividing by 9.x = 54 ÷ 9 = 6
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original Ana: 4 × 6 = £24
Original Bo: 5 × 6 = £30 - Add to find the total. The question asks for the total.£24 + £30 = £54
Check. After the gift: Ana = £21 and Bo = £33. £21 : £33 simplifies (÷ 3) to 7 : 11, which is the ratio 7 : 11 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 4 : 7 means the pens is 4 equal parts and the pencils is 7 equal parts. Let one part be x.Pens = 4x
Pencils = 7x - Write expressions for the amounts after the sale. 12 pencils are sold, so subtract 12 from the pencils. The pens do not change.Pens = 4x
Pencils = 7x − 12 - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.4x7x − 12 = 43
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 3 and by (7x − 12).) Write each product in brackets.3 × 4x = 4 × (7x − 12)
- Expand the brackets. Multiply every term inside each bracket by the number outside.12x = 28x − 48
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.12x = 28x − 48
Subtract 12x from both sides: 0 = 16x − 48
Add 48 to both sides: 48 = 16x - Divide both sides by 16. 16x means 16 multiplied by x, so the inverse is dividing by 16.x = 48 ÷ 16 = 3
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original pens: 4 × 3 = 12
Original pencils: 7 × 3 = 21
Check. After the sale: pens = 12 and pencils = 9. 12 : 9 simplifies (÷ 3) to 4 : 3, which is the ratio 4 : 3 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. Cross-multiplying turns a = b over c = d into a × d = b × c.
Method. Write the ratio as a fraction, cross-multiply to remove the fractions, expand and solve the linear equation.
- Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.x + 4x + 10 = 23
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 3 and by (x + 10).) Write each product in brackets.3 × (x + 4) = 2 × (x + 10)
- Expand the brackets. Multiply every term inside each bracket by the number outside.3x + 12 = 2x + 20
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.3x + 12 = 2x + 20
Subtract 2x from both sides: x + 12 = 20
Subtract 12 from both sides: x = 8 - Divide both sides by 1. 1x means 1 multiplied by x, so the inverse is dividing by 1.x = 8 ÷ 1 = 8
Check. Substitute x = 8: x + 4 = 12 and x + 10 = 18. 12 : 18 = 2 : 3 ✓
Watch out. Cross-multiply carefully: top of one fraction times bottom of the other. Put the expressions in brackets before you multiply.
Key words. An equation says two things are equal. Cross-multiplying turns a = b over c = d into a × d = b × c.
Method. Write the ratio as a fraction, cross-multiply to remove the fractions, expand and solve the linear equation.
- Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.3x + 12x − 1 = 53
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 3 and by (2x − 1).) Write each product in brackets.3 × (3x + 1) = 5 × (2x − 1)
- Expand the brackets. Multiply every term inside each bracket by the number outside.9x + 3 = 10x − 5
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.9x + 3 = 10x − 5
Subtract 9x from both sides: 3 = x − 5
Add 5 to both sides: 8 = x - Divide both sides by 1. 1x means 1 multiplied by x, so the inverse is dividing by 1.x = 8 ÷ 1 = 8
Check. Substitute x = 8: 3x + 1 = 25 and 2x − 1 = 15. 25 : 15 = 5 : 3 ✓
Watch out. Cross-multiply carefully: top of one fraction times bottom of the other. Put the expressions in brackets before you multiply.
Key words. An equation says two things are equal. Cross-multiplying turns a = b over c = d into a × d = b × c.
Method. Write the ratio as a fraction, cross-multiply to remove the fractions, expand and solve the linear equation.
- Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.4x − 2x + 7 = 21
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 1 and by (x + 7).) Write each product in brackets.1 × (4x − 2) = 2 × (x + 7)
- Expand the brackets. Multiply every term inside each bracket by the number outside.4x − 2 = 2x + 14
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.4x − 2 = 2x + 14
Subtract 2x from both sides: 2x − 2 = 14
Add 2 to both sides: 2x = 16 - Divide both sides by 2. 2x means 2 multiplied by x, so the inverse is dividing by 2.x = 16 ÷ 2 = 8
Check. Substitute x = 8: 4x − 2 = 30 and x + 7 = 15. 30 : 15 = 2 : 1 ✓
Watch out. Cross-multiply carefully: top of one fraction times bottom of the other. Put the expressions in brackets before you multiply.
Key words. An equation says two things are equal. Cross-multiplying turns a = b over c = d into a × d = b × c.
Method. Write the ratio as a fraction, cross-multiply to remove the fractions, expand and solve the linear equation.
- Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.2x + 55x − 3 = 1317
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 17 and by (5x − 3).) Write each product in brackets.17 × (2x + 5) = 13 × (5x − 3)
- Expand the brackets. Multiply every term inside each bracket by the number outside.34x + 85 = 65x − 39
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.34x + 85 = 65x − 39
Subtract 34x from both sides: 85 = 31x − 39
Add 39 to both sides: 124 = 31x - Divide both sides by 31. 31x means 31 multiplied by x, so the inverse is dividing by 31.x = 124 ÷ 31 = 4
Check. Substitute x = 4: 2x + 5 = 13 and 5x − 3 = 17. 13 : 17 ✓
Watch out. Cross-multiply carefully: top of one fraction times bottom of the other. Put the expressions in brackets before you multiply.
Key words. An equation says two things are equal. Cross-multiplying turns a = b over c = d into a × d = b × c.
Method. Write the ratio as a fraction, cross-multiply to remove the fractions, expand and solve the linear equation.
- Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.3x − 1x + 9 = 11
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 1 and by (x + 9).) Write each product in brackets.1 × (3x − 1) = 1 × (x + 9)
- Expand the brackets. Multiply every term inside each bracket by the number outside.3x − 1 = x + 9
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.3x − 1 = x + 9
Subtract x from both sides: 2x − 1 = 9
Add 1 to both sides: 2x = 10 - Divide both sides by 2. 2x means 2 multiplied by x, so the inverse is dividing by 2.x = 10 ÷ 2 = 5
Check. Substitute x = 5: 3x − 1 = 14 and x + 9 = 14. 14 : 14 = 1 : 1 ✓
Watch out. Cross-multiply carefully: top of one fraction times bottom of the other. Put the expressions in brackets before you multiply.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Write x and y as multiples of one letter k, substitute, then cancel k.
- Write x and y using one letter. The ratio 3 : 5 means x is 3 parts and y is 5 parts. If one part is k, then x = 3k and y = 5k.x = 3k
y = 5k - Substitute into the first expression. Replace x and y by their values in k.2x + y = 2(3k) + (1)(5k) = 6k + (5)k = 11k
- Substitute into the second expression. Do the same for the second expression.x + 3y = 1(3k) + (3)(5k) = 3k + (15)k = 18k
- Write the ratio and cancel k. Both terms contain k, so k divides out. The ratio of the numbers is what remains.11k : 18k = 11 : 18
Check. Choose numbers in the ratio, e.g. x = 9, y = 15: 2x + y = 33 and x + 3y = 54; the ratio 33 : 54 simplifies to 11 : 18. ✓
Watch out. Cancel k only at the end, when it is a factor of both parts.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Write x and y as multiples of one letter k, substitute, then cancel k.
- Write x and y using one letter. The ratio 4 : 3 means x is 4 parts and y is 3 parts. If one part is k, then x = 4k and y = 3k.x = 4k
y = 3k - Substitute into the first expression. Replace x and y by their values in k.x + y = 1(4k) + (1)(3k) = 4k + (3)k = 7k
- Substitute into the second expression. Do the same for the second expression.x − y = 1(4k) + (-1)(3k) = 4k + (-3)k = 1k
- Write the ratio and cancel k. Both terms contain k, so k divides out. The ratio of the numbers is what remains.7k : 1k = 7 : 1
Check. Choose numbers in the ratio, e.g. x = 12, y = 9: x + y = 21 and x − y = 3; the ratio 21 : 3 simplifies to 7 : 1. ✓
Watch out. Cancel k only at the end, when it is a factor of both parts.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Write x and y as multiples of one letter k, substitute, then cancel k.
- Write x and y using one letter. The ratio 2 : 5 means x is 2 parts and y is 5 parts. If one part is k, then x = 2k and y = 5k.x = 2k
y = 5k - Substitute into the first expression. Replace x and y by their values in k.3x + 2y = 3(2k) + (2)(5k) = 6k + (10)k = 16k
- Substitute into the second expression. Do the same for the second expression.x + y = 1(2k) + (1)(5k) = 2k + (5)k = 7k
- Write the ratio and cancel k. Both terms contain k, so k divides out. The ratio of the numbers is what remains.16k : 7k = 16 : 7
Check. Choose numbers in the ratio, e.g. x = 6, y = 15: 3x + 2y = 48 and x + y = 21; the ratio 48 : 21 simplifies to 16 : 7. ✓
Watch out. Cancel k only at the end, when it is a factor of both parts.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Write x and y as multiples of one letter k, substitute, then cancel k.
- Write x and y using one letter. The ratio 2 : 3 means x is 2 parts and y is 3 parts. If one part is k, then x = 2k and y = 3k.x = 2k
y = 3k - Substitute into the first expression. Replace x and y by their values in k.5x − y = 5(2k) + (-1)(3k) = 10k + (-3)k = 7k
- Substitute into the second expression. Do the same for the second expression.x + 2y = 1(2k) + (2)(3k) = 2k + (6)k = 8k
- Write the ratio and cancel k. Both terms contain k, so k divides out. The ratio of the numbers is what remains.7k : 8k = 7 : 8
Check. Choose numbers in the ratio, e.g. x = 6, y = 9: 5x − y = 21 and x + 2y = 24; the ratio 21 : 24 simplifies to 7 : 8. ✓
Watch out. Cancel k only at the end, when it is a factor of both parts.
Method. Combine the ratios to get a : b : c, then pick out the first and last numbers.
- Spot the quantity that is in both ratios. b appears in both ratios, so it links them together. To combine, b must stand for the same number of parts in both.a : b = 2 : 3
b : c = 4 : 7 - Find the lowest common multiple (LCM) of the two b numbers. We make both b numbers equal to the LCM of 3 and 4.Multiples of 3: 3, 6, 9, 12
Multiples of 4: 4, 8, 12
LCM = 12 - Multiply each ratio so that b becomes 12. Multiplying every part of a ratio by the same number gives an equivalent ratio.a : b = 2 : 3 (multiply by 4) = 8 : 12
b : c = 4 : 7 (multiply by 3) = 12 : 21 - Write the combined ratio a : b : c. b is now the same in both, so the three numbers can be joined.a : b : c = 8 : 12 : 21
- Pick out a : c. We want only the first and last parts of the combined ratio. The middle part b is not needed.a : c = 8 : 21
Check. If a = 8, b = 12, c = 21: a : b = 8 : 12 = 2 : 3 and b : c = 12 : 21 = 4 : 7. ✓
Watch out. Check that a : c does not need simplifying (8 and 21 have no common factor).
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 3 : 5 means the Ali is 3 equal parts and the Bea is 5 equal parts. Let one part be x.Ali = 3x
Bea = 5x - Write expressions for the amounts in 6 years. In 6 years each person is 6 years older, so add 6 to both ages.Ali = 3x + 6
Bea = 5x + 6 - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.3x + 65x + 6 = 57
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 7 and by (5x + 6).) Write each product in brackets.7 × (3x + 6) = 5 × (5x + 6)
- Expand the brackets. Multiply every term inside each bracket by the number outside.21x + 42 = 25x + 30
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.21x + 42 = 25x + 30
Subtract 21x from both sides: 42 = 4x + 30
Subtract 30 from both sides: 12 = 4x - Divide both sides by 4. 4x means 4 multiplied by x, so the inverse is dividing by 4.x = 12 ÷ 4 = 3
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original Ali: 3 × 3 = 9 years
Original Bea: 5 × 3 = 15 years
Check. In 6 years: Ali = 15 years and Bea = 21 years. 15 years : 21 years simplifies (÷ 3) to 5 : 7, which is the ratio 5 : 7 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 3 : 5 means the shorter side is 3 equal parts and the longer side is 5 equal parts. Let one part be x.Shorter side = 3x
Longer side = 5x - Write expressions for the amounts after the change. The shorter side is increased by 4 cm and the longer side is decreased by 2 cm. The new shape is a square, so the sides are equal (ratio 1 : 1).Shorter side = 3x + 4
Longer side = 5x − 2 - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.3x + 45x − 2 = 11
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 1 and by (5x − 2).) Write each product in brackets.1 × (3x + 4) = 1 × (5x − 2)
- Expand the brackets. Multiply every term inside each bracket by the number outside.3x + 4 = 5x − 2
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.3x + 4 = 5x − 2
Subtract 3x from both sides: 4 = 2x − 2
Add 2 to both sides: 6 = 2x - Divide both sides by 2. 2x means 2 multiplied by x, so the inverse is dividing by 2.x = 6 ÷ 2 = 3
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original shorter side: 3 × 3 = 9 cm
Original longer side: 5 × 3 = 15 cm
Check. After the change: shorter side = 13 cm and longer side = 13 cm. 13 cm : 13 cm simplifies (÷ 13) to 1 : 1, which is the ratio 1 : 1 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Find how much of each liquid is in each mixture, add the amounts, then write the ratio of the totals. You cannot just add the ratios.
- Container A: find the amount of each liquid. Container A has 30 litres in the ratio 1 : 4, which is 5 equal parts.1 part = 30 ÷ 5 = 6 litres
Squash: 1 × 6 = 6 litres
Water: 4 × 6 = 24 litres - Container B: find the amount of each liquid. Container B has 20 litres in the ratio 2 : 3, which is 5 equal parts.1 part = 20 ÷ 5 = 4 litres
Squash: 2 × 4 = 8 litres
Water: 3 × 4 = 12 litres - Add the amounts of each liquid. When the mixtures are poured together, the amounts of each liquid add.Squash: 6 + 8 = 14 litres
Water: 24 + 12 = 36 litres - Write the ratio squash : water. Use the totals.14 : 36
- Simplify. Divide both parts by 2.7 : 18
Check. The total volume is 30 + 20 = 50 litres, and 14 + 36 = 50 litres. ✓
Watch out. Never add ratios 1 : 3 and 2 : 1 to get 3 : 4. Work with actual amounts, then form the ratio.
Method. Write a and b using one letter k, substitute into both expressions, then cancel k.
- Write a and b using k. The ratio 3 : 4 means a is 3 parts and b is 4 parts, and one part is k.a = 3k, b = 4k
- Substitute into a + b. Replace a and b.a + b = 3k + 4k = 7k
- Substitute into b − a. Replace a and b.b − a = 4k − 3k = 1k
- Write the ratio and cancel k. Both parts have the factor k.7k : k = 7 : 1
Check. Try a = 6 and b = 8: a + b = 14 and b − a = 2, and 14 : 2 = 7 : 1. ✓
Watch out. Always say what each letter stands for, and write the final line as the result you were asked to show.
Method. Write the amounts as 2x and 3x, add 5 to each, set the new ratio equal to 2 : 3 and see whether any x works.
- Write the amounts. The ratio 2 : 3 means 2x and 3x.2x and 3x
- Add 5 to each. New amounts.2x + 5 and 3x + 5
- Set up the equation for the new ratio 2 : 3. Cross-multiply: 3 times the first expression equals 2 times the second.3(2x + 5) = 2(3x + 5)
- Expand both sides. Multiply each term inside the bracket.6x + 15 = 6x + 10
- Subtract 6x from both sides. The x terms cancel.15 = 10
- Conclude. 15 is not equal to 10, so no value of x works. Adding the same number to both parts always changes the ratio.It is never possible.
Check. Try x = 1: 2 : 3 becomes 7 : 8, which is not 2 : 3. Try x = 10: 20 : 30 becomes 25 : 35 = 5 : 7, not 2 : 3. ✓
Watch out. To keep a ratio the same, multiply or divide both parts. Adding changes it.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 3 : 5 means the red is 3 equal parts and the blue is 5 equal parts. Let one part be x.Red = 3x
Blue = 5x - Write expressions for the amounts after the change. Eight red marbles are removed, so subtract 8 from the red. Twelve blue marbles are added, so add 12 to the blue.Red = 3x − 8
Blue = 5x + 12 - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.3x − 85x + 12 = 13
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 3 and by (5x + 12).) Write each product in brackets.3 × (3x − 8) = 1 × (5x + 12)
- Expand the brackets. Multiply every term inside each bracket by the number outside.9x − 24 = 5x + 12
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.9x − 24 = 5x + 12
Subtract 5x from both sides: 4x − 24 = 12
Add 24 to both sides: 4x = 36 - Divide both sides by 4. 4x means 4 multiplied by x, so the inverse is dividing by 4.x = 36 ÷ 4 = 9
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original red: 3 × 9 = 27
Original blue: 5 × 9 = 45 - Add to find the total. The question asks for the total.27 + 45 = 72
Check. After the change: red = 19 and blue = 57. 19 : 57 simplifies (÷ 19) to 1 : 3, which is the ratio 1 : 3 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. An equation says two things are equal. We form an equation from the information in the question, then solve it to find x. Cross-multiplying means multiplying the top of each fraction by the bottom of the other.
Method. Write the amounts as multiples of one letter (the value of one part). Use the change to write new expressions, then form and solve an equation from the new ratio.
- Use a letter for the value of one part. The ratio 3 : 2 means the girls is 3 equal parts and the boys is 2 equal parts. Let one part be x.Girls = 3x
Boys = 2x - Write expressions for the amounts after the change. Seven girls leave, so subtract 7 from the girls. The numbers are equal now, so the ratio is 1 : 1.Girls = 3x − 7
Boys = 2x - Write the ratio as a fraction (or use cross-multiplying). Equal ratios give equal fractions: the first quantity divided by the second equals r over s. Then we can multiply both sides to remove the fractions.3x − 72x = 11
- Cross-multiply. Multiply the top of each fraction by the bottom of the other. (This is the same as multiplying both sides by 1 and by 2x.) Write each product in brackets.1 × (3x − 7) = 1 × 2x
- Expand the brackets. Multiply every term inside each bracket by the number outside.3x − 7 = 2x
- Collect the x terms on one side and the numbers on the other. Do the same to both sides so that the equation stays balanced.3x − 7 = 2x
Subtract 2x from both sides: x − 7 = 0
Add 7 to both sides: x = 7 - Divide both sides by 1. 1x means 1 multiplied by x, so the inverse is dividing by 1.x = 7 ÷ 1 = 7
- Find the amounts the question asks for. Put the value of one part back into the expressions.Original girls: 3 × 7 = 21
Original boys: 2 × 7 = 14
Check. After the change: girls = 14 and boys = 14. 14 : 14 simplifies (÷ 14) to 1 : 1, which is the ratio 1 : 1 we were told. ✓
Watch out. The ratio is only true for the amounts after the change. Before the change the amounts are in the original ratio. Do not mix the two.
Key words. A ratio compares the sizes of two or more quantities, written with a colon, for example 2 : 3 (read “2 to 3”). The numbers in a ratio are called parts. The simplest form of a ratio has no whole number (other than 1) that divides into every part.
Method. Form and solve an equation to find A and B. Then compare the sizes of the ratios as fractions.
- Write A and B using x. The ratio 3 : 4 means A = 3x and B = 4x.A = 3x, B = 4x
- Add 12 to each and form the equation. The new ratio is 5 : 6, so cross-multiply: 6(3x + 12) = 5(4x + 12).6(3x + 12) = 5(4x + 12)
- Expand. Multiply out each bracket.18x + 72 = 20x + 60
- Collect terms. Subtract 18x and 60 from both sides.12 = 2x
- Solve. Divide both sides by 2.x = 6
- Find A and B. Substitute x = 6.A = 3 × 6 = 18 and B = 4 × 6 = 24
- Compare the ratios as fractions. The closer a ratio is to 1 : 1, the closer the fraction smaller ÷ larger is to 1.Old: 18 ÷ 24 = 0.75. New: 30 ÷ 36 = 0.8333... 0.8333... is closer to 1.
Check. New amounts: 18 + 12 = 30 and 24 + 12 = 36, and 30 : 36 = 5 : 6. ✓
Watch out. Adding the same amount to both parts always moves the ratio closer to 1 : 1.