Year 6: Ratio, proportion and algebra
Part of the Year 6 sats maths revision chapter. Chapter overview · whole chapter on one page
Ratio, proportion and algebra
A ratio compares two amounts. The ratio 2:3 means for every 2 parts of one thing there are 3 parts of the other. To share an amount in a ratio, first find how many parts there are altogether, then share the total into one part.
A formula is a rule written with letters. Put the number in place of the letter, then work out the answer, multiplying before adding. In a linear sequence the terms go up by the same amount each time.
An equation such as 3n + 7 = 31 has one answer. Undo the addition, then the multiplication. Do the same to both sides.
Key ideas
- A ratio shares an amount: add the parts, find one part, then multiply.
- A formula is worked out by putting the number in place of the letter, then multiplying before adding.
- The nth term is a rule for every term: 4n + 3 gives 7, 11, 15, and so on.
- To solve an equation, do the same to both sides until the letter is on its own.
Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.
Method. Add the parts, share the total into one part, then multiply each share by its number of parts.
- Add the parts of the ratio. The ratio 3:4 has 7 equal parts altogether.3 + 4 = 7
- Find the value of one part. Share the total equally between all the parts.£84 ÷ 7 = £12
- Multiply each share by its number of parts. The first share has 3 parts and the second has 4 parts.3 × £12 = £36, and 4 × £12 = £48
- Write the answer. £36 and £48
Check. £36 + £48 = £84, so the shares add up to the total. ✓
Watch out. Share the total into all the parts, not just between two shares.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. Put the number in place of the letter, then multiply before you add.
- Write the formula with n = 4. Put the number in place of the letter n.6n + 2 with n = 4
- Multiply first. Multiplication comes before addition.6 × 4 = 24
- Add the number. Add the last number to finish the calculation.24 + 2 = 26
Check. 26 − 2 = 24, and 24 ÷ 6 = 4, so the answer is right. ✓
Watch out. Multiply before you add: 3n + 4 with n = 5 is 15 + 4 = 19, not 35.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. Undo the addition, then undo the multiplication. Do the same to both sides.
- Take 5 away from both sides. Undo the addition first.4n = 29 − 5 = 24
- Divide both sides by 4. Undo the multiplication to find n.n = 24 ÷ 4 = 6
- Check the answer. Put n back into the first equation.4 × 6 + 5 = 29
Check. 4 × 6 + 5 = 29, which is the equation, so n = 6 is right. ✓
Watch out. Do the same thing to both sides. Undo the addition before the multiplication.
Watch out
Common mistake: in a formula, adding before multiplying. In 6n + 2 with n = 4, first do 6 × 4 = 24, then add 2, to get 26. Do not do 4 + 2 first.
Exercise 13D Timed set: ratio, formulae and sequences (about 12 minutes)
Non-calculatorKey words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.
Method. Add the parts, share the total into one part, then multiply each share by its number of parts.
- Add the parts of the ratio. The ratio 2:3 has 5 equal parts altogether.2 + 3 = 5
- Find the value of one part. Share the total equally between all the parts.£60 ÷ 5 = £12
- Multiply each share by its number of parts. The first share has 2 parts and the second has 3 parts.2 × £12 = £24, and 3 × £12 = £36
- Write the answer. £24 and £36
Check. £24 + £36 = £60, so the shares add up to the total. ✓
Watch out. Share the total into all the parts, not just between two shares.
Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.
Method. Divide the known amount by its parts to find one part, then multiply one part by the other number of parts.
- Find the value of one part. The first amount is 3 parts, so divide it by 3.18 ÷ 3 = 6
- Find the second amount. The second amount is 5 parts, so multiply the value of one part by 5.5 × 6 = 30
Check. The two amounts keep the ratio 3:5, because 18 : 30 = 3:5. ✓
Watch out. Find one part first. Do not just multiply the known amount by the ratio.
Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.
Method. Find the cost of one item, then multiply by the number of items.
- Find the cost of one item. 5 items cost £3.20, so share the cost equally.£3.20 ÷ 5 = £0.64
- Multiply by 8. Each item costs the same, so multiply the cost of one item by 8.£0.64 × 8 = £5.12
Check. £5.12 ÷ 8 = £0.64, so the cost of one item is right. ✓
Watch out. Find the cost of one item first. Do not multiply the price of the first group by the number of items you want.
Key words. A ratio compares two amounts, such as 2:3. A part is one of the equal shares in the ratio. A scale factor is the number you multiply by to make a shape bigger or smaller.
Method. Find the scale factor, then multiply the other length by it.
- Find the scale factor. The scale factor is the new length divided by the old length.10 ÷ 4 = 2.5
- Multiply the other side by the scale factor. Every length in the new shape is the old length times the same scale factor.6 × 2.5 = 15
Check. 15 ÷ 2.5 = 6, which is the old length. ✓
Watch out. Use the same scale factor for every length. Do not add the same amount to each side.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. Put the number in place of the letter, then multiply before you add.
- Write the formula with n = 7. Put the number in place of the letter n.5n + 3 with n = 7
- Multiply first. Multiplication comes before addition.5 × 7 = 35
- Add the number. Add the last number to finish the calculation.35 + 3 = 38
Check. 38 − 3 = 35, and 35 ÷ 5 = 7, so the answer is right. ✓
Watch out. Multiply before you add: 3n + 4 with n = 5 is 15 + 4 = 19, not 35.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. Substitute the length and the width into the formula, working out the brackets first.
- Write the formula. The perimeter of a rectangle is two lengths plus two widths.P = 2 × (l + w)
- Add the length and the width. Work out the brackets first.8 + 5 = 13
- Multiply by 2. There are two lengths and two widths.2 × 13 = 26
Check. 26 is the same as adding 8, 5, 8 and 5. ✓
Watch out. Work out the brackets first, then multiply by 2.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. Find how much the terms go up by, then find the number to add so that the first term is right.
- Find the difference between terms. Subtract each term from the one after it.7, 12, 17, 22: each step adds 5
- Write the part with n. The terms go up by 5 each time, so the rule has 5n.5n
- Find the number to add. When n = 1 the term is 7, and 5 × 1 = 5, so add 2.5 × 1 + 2 = 7
- Write the rule. 5n + 2
Check. Test n = 2: 5 × 2 + 2 = 12, which is the second term. ✓
Watch out. Check the rule with the second term. Then the rule must work for every term.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. Put the number in place of the letter, then multiply before you add.
- Write the formula with n = 10. Put the number in place of the letter n.4n + 3 with n = 10
- Multiply first. Multiplication comes before addition.4 × 10 = 40
- Add the number. Add the last number to finish the calculation.40 + 3 = 43
Check. 43 − 3 = 40, and 40 ÷ 4 = 10, so the answer is right. ✓
Watch out. Multiply before you add: 3n + 4 with n = 5 is 15 + 4 = 19, not 35.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. Undo the addition, then undo the multiplication. Do the same to both sides.
- Take 7 away from both sides. Undo the addition first.3n = 31 − 7 = 24
- Divide both sides by 3. Undo the multiplication to find n.n = 24 ÷ 3 = 8
- Check the answer. Put n back into the first equation.3 × 8 + 7 = 31
Check. 3 × 8 + 7 = 31, which is the equation, so n = 8 is right. ✓
Watch out. Do the same thing to both sides. Undo the addition before the multiplication.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. Replace one letter so that there is only one unknown, then solve.
- Write the two facts. One number is a, the other is b.a + b = 20 and a = 3b
- Put the first fact into the second. Replace a with 3b in the sum.3b + b = 20, so 4b = 20
- Divide by 4. Find the value of b.b = 20 ÷ 4 = 5
- Find a. a is 3 times b.a = 3 × 5 = 15
- Write the answer. 15 and 5
Check. 15 + 5 = 20, and 15 is 3 times 5. ✓
Watch out. Replace a letter in one equation with the other. Then there is only one unknown.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. List every pair of whole numbers that adds to the total, and count each pair only once.
- List the pairs, starting with the smallest number. Start with 1 and the biggest partner, and move towards the middle.1 + 11, 2 + 10, 3 + 9, 4 + 8, 5 + 7, 6 + 6
- Count the pairs. Each pair is counted once, because order does not matter.6 pairs
- Write the answer. 6
Check. Every pair adds to 12, and no pair is missing from the list. ✓
Watch out. Count each pair once: 2 + 10 and 10 + 2 are the same pair.
Key words. A formula is a rule written with letters. To substitute means to put in a number in place of a letter. A term is one number in a sequence, and the term-to-term rule says how to get from one term to the next.
Method. Work out the 10th term yourself and compare it with what Sam says.
- Work out the term. Put n = 10 into the rule 3n + 2.3 × 10 + 2 = 30 + 2
- Add. Add the number to the answer.30 + 2 = 32
- Compare with Sam. Sam says the answer is 30, which is not 32.No, it is 32
Check. The first two terms are 5 and 8, which go up by 3 each time, so the rule is right. ✓
Watch out. Check Sam's claim by working out the term. Do not guess.