GCSE Maths Number Basics (Higher): check-up and exam-style questions

Key words. Place value means the value of a digit because of the column it is in. Units (ones) are the whole numbers; tenths and hundredths are the parts after the decimal point.

Method. The value of a digit depends on its column. Find the column, then multiply the digit by what that column is worth.

1. Write the number in a place value table. Each column is worth ten times the column on its right. The digits sit in columns, and the decimal point separates the whole numbers from the parts.

   tens	units		tenths	hundredths	thousandths
   2	9	.	0	5	6

2. Find the column of the digit 5. Reading the headings, the 5 is in the hundredths column. One in this column is worth 0.01.
   The hundredths column is worth 0.01.
3. Multiply the digit by the column value. There are 5 lots of 0.01.
   5 × 0.01 = 0.05

Check. In words, the digit 5 stands for 5 hundredths, which is 0.05. ✓

Watch out. A digit is worth its column value, not just the digit itself. The 7 in 4 700 is worth 700, not 7.

Method. Make the numbers easy to compare, then order them smallest to largest.

1. Write every number with the same number of decimal places, using zeros. The extra zeros do not change the values. They make the columns line up so we can compare digit by digit.
   0.38 = 0.380
   0.308 = 0.308
   0.83 = 0.830
   0.3 = 0.300
2. Compare the sizes, ignoring the decimal point and any minus signs. Now every number has 3 decimal places, so we can compare them as if they were whole numbers.
   0.38: 380, 0.308: 308, 0.83: 830, 0.3: 300
3. Write the original numbers in order. Order smallest to largest.
   0.3, 0.308, 0.38, 0.83

Check. Check each neighbouring pair: 0.3 < 0.308; 0.308 < 0.38; 0.38 < 0.83. ✓

Watch out. Ordering decimals: line up the decimal places. Do not decide by the length of the number.

Key words. The sum or total is the answer to an addition. Carry means move the tens digit of a column total to the next column on the left.

Method. Add using columns. Line up the place values, add from the right, and carry when a column reaches 10 or more.

1. Line up the numbers in columns. Digits can only be added when they are worth the same, so units go under units, tens under tens, and decimal points under decimal points. Fill any gaps with zeros.
   48.70 + 5.36 -------
2. Add the columns, starting on the right. We start with the smallest column. If a column adds up to 10 or more, we write the units digit and carry the tens digit to the next column.
   Hundredths: 0 + 6 = 6. Write 6.
   Tenths: 7 + 3 = 10. Write 0 and carry 1 into the units column.
   Units: 8 + 5 + 1 (carried) = 14. Write 4 and carry 1 into the tens column.
   Tens: 4 + 0 + 1 (carried) = 5. Write 5.
3. Put the decimal point in the answer. The point goes directly under the other decimal points.
   48.70 + 5.36 ------- 54.06 11
   48.7 + 5.36 = 54.06

Check. Check with an inverse (opposite) operation: 54.06 − 5.36 = 48.7, which is the first number. ✓

Watch out. Line up the decimal points, not the right-hand ends of the numbers. 5.2 + 0.37 needs 5.20 + 0.37.

Key words. The difference is the answer to a subtraction. Exchange (or borrowing) means taking 1 from a column and putting 10 into the column on its right.

Method. Subtract using columns. Line up the place values, subtract from the right, and exchange when the top digit is too small.

1. Line up the numbers in columns, with the bigger number on top. Digits can only be subtracted when they are worth the same: units under units, tens under tens.
   7002 − 3458 ------
2. Subtract each column, starting on the right. If the top digit is smaller than the digit below it, we exchange (borrow): take 1 from the column on the left and add it to this column as 10.
   Units: 2 is smaller than 8, so we must exchange. The tens and hundreds columns are 0, so there is nothing to take there. Go left to the thousands column. Take 1 thousand (7 becomes 6) and exchange it for 10 hundreds. Take 1 of those hundreds (leaving 9) and exchange it for 10 tens. Take 1 of those tens (leaving 9) and exchange it for 10 units. 2 becomes 12. Now 12 − 8 = 4.
   Tens: 9 − 5 = 4.
   Hundreds: 9 − 4 = 5.
   Thousands: 6 − 3 = 3.
3. Write the answer. Read the bottom row.
   7002 − 3458 ------ 3544
   7002 − 3458 = 3544

Check. Check with the inverse (opposite) operation, addition: 3544 + 3458 = 7002. ✓

Watch out. Always subtract the smaller number from the bigger in each column only after exchanging. Never take the top digit from the bottom one.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 86 on top and 47 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   86 × 47 ----
2. Multiply 86 by the units digit, 7. Work from the right. Multiply each digit of the top number by 7. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   6 × 7 = 42: write 2, carry 4
   8 × 7 + 4 (carried) = 60: write 0, carry 6
   Write the last carry: 6
   So 86 × 7 = 602.
3. Multiply 86 by the tens digit, 4 (which means 40). The 4 is in the tens column, so it is really 40. First write 1 zero on the right as place holder, then multiply by 4.
   6 × 4 = 24: write 4, carry 2
   8 × 4 + 2 (carried) = 34: write 4, carry 3
   Write the last carry: 3
   86 × 4 = 344, then add 1 zero: 86 × 40 = 3440.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   602 + 3440 = 4042
   86 × 47 ---- 602 3440 ----- 4042

Check. Estimate by rounding each number to 1 significant figure: 90 × 50 = 4500. This is close to 4042, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Long division: work from the left, dividing 22 into each number in turn, carrying the remainder to the next digit.

1. Write out the 22 times table up to 22 × 9. Having the multiples ready makes it quick to see how many times 22 goes into each number.
   22 × 1 = 22
   22 × 2 = 44
   22 × 3 = 66
   22 × 4 = 88
   22 × 5 = 110
   22 × 6 = 132
   22 × 7 = 154
   22 × 8 = 176
   22 × 9 = 198
2. Set out the division. Write 1496 under the "bus stop" and 22 outside it. The answer will be written on top, one digit above each digit we divide.
   22 ) 1496
3. Start with the first 3 digits, 149. 22 does not go into the first digit, so we take digits from the left until we have a number that 22 can go into.
   22 goes into 149 6 times: 22 × 6 = 132
   Subtract: 149 − 132 = 17
4. Bring down the next digit, 6, to make 176. The remainder 17 goes in front of the 6. This is the same as carrying the remainder to the next column.
   22 goes into 176 8 times: 22 × 8 = 176
   Subtract: 176 − 176 = 0
5. Write the answer. Read the digits along the top.
   68 22) 1496 132 --- 176 176 --- 0
   1496 ÷ 22 = 68

Check. Multiply back: 68 × 22 = 1496. ✓

Watch out. If 22 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).

Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.

1. Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.
   −4 − (−9)
2. Deal with the two signs next to each other: subtracting a negative. Subtracting a negative cancels it out: if someone takes away a debt of 9, you are 9 better off. So "− (−9)" is the same as "+ 9".
   −4 − (−9) = −4 + 9
3. Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.
   Start at −4 on the number line and move 9 steps to the right (adding moves up).
   4 steps take you to 0, then 5 more steps take you to 5.
   −4 + 9 = 5

Check. Reverse the calculation: 5 + (−9) = −4. ✓

Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.

Key words. BIDMAS is the order of priority in a calculation: Brackets, Indices (powers and roots), Division and Multiplication (same priority, left to right), Addition and Subtraction (same priority, left to right).

Method. Follow BIDMAS: brackets first, then indices (powers and roots), then division and multiplication, then addition and subtraction. Work from left to right within each level.

1. Write the calculation and remember the order. Calculations are not simply done from left to right. The agreed order is BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
   24 ÷ (2 + 4) + 3²
2. Brackets first. Add: 2 + 4. Only addition or subtraction is left, so do it now.
   24 ÷ (2 + 4) + 3²
   2 + 4 = 6
   = 24 ÷ 6 + 3²
3. Indices (powers): work out 3². A power tells us how many times to multiply a number by itself. After brackets, powers and roots come next.
   24 ÷ 6 + 3²
   3² = 3 × 3 = 9
   = 24 ÷ 6 + 9
4. Divide: 24 ÷ 6. Multiplication and division are done before addition and subtraction.
   24 ÷ 6 + 9
   24 ÷ 6 = 4
   = 4 + 9
5. Add: 4 + 9. Only addition or subtraction is left, so do it now.
   4 + 9
   4 + 9 = 13
   = 13

Check. A scientific calculator follows the same order (BIDMAS), so typing in the whole calculation gives 13. ✓

Watch out. Do not work from left to right just because it looks easier. In 2 + 3 × 4 the multiplication comes first, so the answer is 14, not 20.

Key words. A unit says what we measure in. Metric units are based on 10, 100 and 1000, so converting means multiplying or dividing by powers of 10.

Method. To change units, use the conversion fact. Change to a smaller unit: multiply. Change to a bigger unit: divide.

1. Write down the conversion fact. We need to know how many of the smaller unit make one of the bigger unit.
   1 km = 1000 m
2. Decide: km to m means changing to a smaller unit, so multiply. A smaller unit means you need more of them. For example 2 km is 2000 m. So we multiply by 1000.
   3.4 km × 1000
3. Calculate.
   3.4 × 1000 = 3400
4. Write the answer with its unit. Always put the new unit in the answer.
   3.4 km = 3400 m

Check. Convert back: 3400 ÷ 1000 = 3.4. ✓

Watch out. Check the direction: there are always more of the smaller unit. If your answer for centimetres is smaller than the number of metres, you divided when you should have multiplied.

Key words. The reciprocal of a number is the number you multiply it by to get 1. Dividing by a number is the same as multiplying by its reciprocal.

Method. To find a reciprocal, write the number as a fraction and turn it upside down (swap the top and the bottom).

1. Remember what a reciprocal is. The reciprocal of a number is 1 divided by that number. Two numbers are reciprocals if their product is 1. For a fraction, the reciprocal is the fraction turned upside down.
   reciprocal of x = 1x
2. Turn the fraction upside down. Swap the numerator (top) and the denominator (bottom).
   34 → 43

Check. Multiply the number by its reciprocal: 34 × 43 = 1. ✓

Watch out. The reciprocal is not the negative of the number. The reciprocal of 4 is 1/4, not −4. And 0 has no reciprocal, because you cannot divide by 0.

Method. Read the number in groups of three digits (millions, thousands, then the rest) and say each group in words.

1. Split the whole number into groups of three digits from the right. Groups are read as millions, thousands and then the last three digits. The gaps in the number show the groups.
   70 thousand | 305
2. Say each group in words. Each group is a number from 0 to 999 followed by its name (million or thousand). Use "and" before the tens and units, e.g. one hundred and five.
   70 thousand = seventy thousand
   305 = three hundred and five
3. Join the groups. Put the groups together in order, with "and" before the last part if it is less than 100.
   seventy thousand three hundred and five

Check. Say it out loud: does "seventy thousand three hundred and five" sound like 70 305? ✓

Watch out. Do not write "and" between every part. We say "four thousand and fifty-six", but "four thousand two hundred and six".

Method. Make the numbers easy to compare, then order them smallest to largest.

1. Write every number with the same number of decimal places, using zeros. The extra zeros do not change the values. They make the columns line up so we can compare digit by digit.
   0.45 = 0.450
   0.5 = 0.500
   0.405 = 0.405
   0.54 = 0.540
2. Compare the sizes, ignoring the decimal point and any minus signs. Now every number has 3 decimal places, so we can compare them as if they were whole numbers.
   0.45: 450, 0.5: 500, 0.405: 405, 0.54: 540
3. Write the original numbers in order. Order smallest to largest.
   0.405, 0.45, 0.5, 0.54

Check. Check each neighbouring pair: 0.405 < 0.45; 0.45 < 0.5; 0.5 < 0.54. ✓

Watch out. Ordering decimals: line up the decimal places. Do not decide by the length of the number.

Key words. The product is the answer to a multiplication. A place holder is a 0 used to keep digits in the correct column.

Method. Long multiplication: multiply by each digit of the second number (units, tens, hundreds...) and add the rows.

1. Set out the multiplication. Write 427 on top and 36 underneath, with the units under the units. We multiply the top number by each digit of the bottom number, then add the rows.
   427 × 36 -----
2. Multiply 427 by the units digit, 6. Work from the right. Multiply each digit of the top number by 6. If the answer is 10 or more, write the units digit and carry the rest into the next column.
   7 × 6 = 42: write 2, carry 4
   2 × 6 + 4 (carried) = 16: write 6, carry 1
   4 × 6 + 1 (carried) = 25: write 5, carry 2
   Write the last carry: 2
   So 427 × 6 = 2562.
3. Multiply 427 by the tens digit, 3 (which means 30). The 3 is in the tens column, so it is really 30. First write 1 zero on the right as place holder, then multiply by 3.
   7 × 3 = 21: write 1, carry 2
   2 × 3 + 2 (carried) = 8: write 8
   4 × 3 = 12: write 2, carry 1
   Write the last carry: 1
   427 × 3 = 1281, then add 1 zero: 427 × 30 = 12 810.
4. Add the rows. Long multiplication splits the multiplier into parts (units, tens, ...). Adding the partial answers gives the whole answer.
   2562 + 12 810 = 15 372
   427 × 36 ----- 2562 12810 ------ 15372

Check. Estimate by rounding each number to 1 significant figure: 400 × 40 = 16 000. This is close to 15 372, so the size is sensible. ✓

Watch out. When you multiply by the tens digit, write the place-holder 0 first. Forgetting it makes the answer ten times too small.

Key words. The quotient is the answer to a division. The remainder is what is left over. The divisor is the number you divide by.

Method. Long division: work from the left, dividing 15 into each number in turn, carrying the remainder to the next digit.

1. Write out the 15 times table up to 15 × 9. Having the multiples ready makes it quick to see how many times 15 goes into each number.
   15 × 1 = 15
   15 × 2 = 30
   15 × 3 = 45
   15 × 4 = 60
   15 × 5 = 75
   15 × 6 = 90
   15 × 7 = 105
   15 × 8 = 120
   15 × 9 = 135
2. Set out the division. Write 1755 under the "bus stop" and 15 outside it. The answer will be written on top, one digit above each digit we divide.
   15 ) 1755
3. Start with the first 2 digits, 17. 15 does not go into the first digit, so we take digits from the left until we have a number that 15 can go into.
   15 goes into 17 1 time: 15 × 1 = 15
   Subtract: 17 − 15 = 2
4. Bring down the next digit, 5, to make 25. The remainder 2 goes in front of the 5. This is the same as carrying the remainder to the next column.
   15 goes into 25 1 time: 15 × 1 = 15
   Subtract: 25 − 15 = 10
5. Bring down the next digit, 5, to make 105. The remainder 10 goes in front of the 5. This is the same as carrying the remainder to the next column.
   15 goes into 105 7 times: 15 × 7 = 105
   Subtract: 105 − 105 = 0
6. Write the answer. Read the digits along the top.
   117 15) 1755 15 -- 25 15 -- 105 105 --- 0
   1755 ÷ 15 = 117

Check. Multiply back: 117 × 15 = 1755. ✓

Watch out. If 15 does not go into a digit, write 0 on top and carry on. Do not skip the 0, or the answer will be too small.

Key words. A negative number is a number less than zero, such as −3 (three below zero). An integer is a whole number: positive, negative or zero. The sign of a number tells us if it is positive (+) or negative (−).

Method. Replace any double signs by a single sign, then use a number line: adding moves right, subtracting moves left.

1. Write the calculation with brackets round any negative number that follows a sign. This stops two signs from sitting next to each other without brackets and getting muddled.
   −7 − (−3)
2. Deal with the two signs next to each other: subtracting a negative. Subtracting a negative cancels it out: if someone takes away a debt of 3, you are 3 better off. So "− (−3)" is the same as "+ 3".
   −7 − (−3) = −7 + 3
3. Now add or subtract the positive number using a number line. Think of a number line: right is bigger, left is smaller.
   Start at −7 on the number line and move 3 steps to the right (adding moves up).
   You are still below zero when you stop, at −4.
   −7 + 3 = −4

Check. Reverse the calculation: −4 + (−3) = −7. ✓

Watch out. Two minus signs together, − (−), make a plus. A plus and a minus together make a minus.

Key words. BIDMAS is the order of priority in a calculation: Brackets, Indices (powers and roots), Division and Multiplication (same priority, left to right), Addition and Subtraction (same priority, left to right).

Method. Follow BIDMAS: brackets first, then indices (powers and roots), then division and multiplication, then addition and subtraction. Work from left to right within each level.

1. Write the calculation and remember the order. Calculations are not simply done from left to right. The agreed order is BIDMAS: Brackets, Indices (powers and roots), Division and Multiplication, Addition and Subtraction.
   5 + 3 × (8 − 2)²
2. Brackets first. Subtract: 8 − 2. Only addition or subtraction is left, so do it now.
   5 + 3 × (8 − 2)²
   8 − 2 = 6
   = 5 + 3 × 6²
3. Indices (powers): work out 6². A power tells us how many times to multiply a number by itself. After brackets, powers and roots come next.
   5 + 3 × 6²
   6² = 6 × 6 = 36
   = 5 + 3 × 36
4. Multiply: 3 × 36. Multiplication and division are done before addition and subtraction.
   5 + 3 × 36
   3 × 36 = 108
   = 5 + 108
5. Add: 5 + 108. Only addition or subtraction is left, so do it now.
   5 + 108
   5 + 108 = 113
   = 113

Check. A scientific calculator follows the same order (BIDMAS), so typing in the whole calculation gives 113. ✓

Watch out. Do not work from left to right just because it looks easier. In 2 + 3 × 4 the multiplication comes first, so the answer is 14, not 20.

Method. Find the cost of the adult tickets and the child tickets, add, then subtract from £50.

1. Cost of the adult tickets. 2 × £8.50 (1 mark for a correct method).
   2 × 8.50 = 17.00
2. Cost of the child tickets. 3 × £5.20.
   3 × 5.20 = 15.60
3. Add to find the total.
   17.00 + 15.60 = 32.60
4. Subtract from £50. Change = amount paid − cost.
   50.00 − 32.60 = £17.40

Check. 32.60 + 17.40 = 50.00. ✓

Watch out. Marks: 1 for each ticket cost, 1 for the total, 1 for the change.

Method. Count on from the start time to the end time.

1. Count on to the next hour. 08:45 to 09:00 is 15 minutes.
   15 minutes
2. Count whole hours. 09:00 to 11:00.
   2 hours
3. Count the remaining minutes and add. 11:00 to 11:20 is 20 minutes.
   15 + 20 = 35 minutes, so 2 hours 35 minutes

Check. 08:45 + 2 h 35 min = 11:20. ✓

Watch out. Marks: 1 for the method of counting on, 1 for the answer.

Method. Change kilograms to grams, then divide by the amount in each bag.

1. Change 2.4 kg into grams. 1 kg = 1000 g (1 mark).
   2.4 × 1000 = 2400 g
2. Divide by 150. The number of bags is the number of lots of 150 g in 2400 g (1 mark).
   2400 ÷ 150 = 16
3. Write the answer. There is no remainder (1 mark).
   16 bags

Check. 16 × 150 = 2400. ✓

Method. Use the given fact. Work out how each number has changed (a power of 10), then change the answer in the same way.

1. Write down the fact you are given. The question tells us this result so that we do not have to calculate from scratch.
   28 × 35 = 980
2. Compare each number in the new calculation with the numbers in the given fact. Look for how many times 10 each number has been multiplied or divided by.
   28 → 2.8: 28 ÷ 10 = 2.8
   35 → 35: no change
3. Change the answer in the same way. Each change to a number changes the answer in the same way. Together, the answer changes by ÷ 10.
   980 ÷ 10 = 98

Check. Rounded check: 3 × 40 = 120, which is close to 98. ✓

Watch out. When a number is multiplied by 10 and another by 100, the answer is multiplied by 10 × 100 = 1000.

Method. Work out how many coaches are needed (round up), then multiply by the cost of one coach.

1. Divide the students by the seats on a coach. 235 ÷ 48 (1 mark).
   235 ÷ 48 = 4 remainder 43
2. Round up. The 43 students left over need another coach (1 mark).
   4 + 1 = 5 coaches
3. Multiply by the cost of one coach. (1 mark for method)
   5 × 185 = £925

Check. 4 coaches hold 192 students, which is not enough; 5 coaches hold 240. ✓

Watch out. Marks: 1 for the division, 1 for rounding up, 1 for the cost.

Method. Multiply the numbers without the decimal points and count the decimal places.

1. Ignore the decimal points.
   3 × 4 = 12
2. Count the decimal places in the question. 0.3 has 1 and 0.04 has 2.
   1 + 2 = 3 decimal places
3. Put the point back. 12 with 3 decimal places is 0.012.
   0.3 × 0.04 = 0.012

Check. 0.012 ÷ 0.04 = 0.3. ✓

Method. Use BIDMAS to find the correct answer, then compare.

1. Work out the correct answer. Multiplication comes before addition.
   6 × 5 = 30, then 4 + 30 = 34
2. Compare with the given working. The working added 4 + 6 first.
   50 ≠ 34, so the working is incorrect.

Check. 34 − 4 = 30 = 6 × 5. ✓

Method. The perimeter is the distance round the edge: 2 × (length + width).

1. Add the length and width. (1 mark)
   6.5 + 4 = 10.5
2. Double it. There are two lengths and two widths (1 mark).
   10.5 × 2 = 21 cm

Check. 6.5 + 4 + 6.5 + 4 = 21. ✓

Method. Multiply both numbers by 100 to make the divisor a whole number, then divide.

1. Multiply both numbers by 100. (1 mark)
   4.68 ÷ 0.18 = 468 ÷ 18
2. Divide. 18 into 46 is 2 (36), remainder 10; bring down 8 to make 108; 18 into 108 is 6 (1 mark).
   468 ÷ 18 = 26

Check. 26 × 0.18 = 4.68. ✓

Method. Follow BIDMAS: the power first, then the multiplication, then the subtraction.

1. Indices. (−3)² = (−3) × (−3) = 9 (1 mark).
   9 − 4 × (−2)
2. Multiplication. A positive times a negative is negative (1 mark).
   4 × (−2) = −8, so 9 − (−8)
3. Subtract. Subtracting a negative is the same as adding (1 mark).
   9 + 8 = 17

Check. 17 − 8 = 9. ✓

Method. Speed = distance ÷ time (in hours). Then use distance = speed × time.

1. Change the time to hours. 30 minutes = 0.5 hours (1 mark).
   1 hour 30 minutes = 1.5 hours
2. Work out the speed. (1 mark)
   135 ÷ 1.5 = 90 km/h
3. Change 2 hours 20 minutes to minutes and find the distance per minute. It is easier to use minutes: 90 km/h is 90 ÷ 60 = 1.5 km per minute (1 mark).
   2 h 20 min = 140 minutes
4. Work out the distance. (1 mark)
   1.5 × 140 = 210 km

Check. 90 km/h for 2 h is 180 km, and for 20 min (one third of an hour) is 30 km: 180 + 30 = 210. ✓

Method. Multiply both numbers by 1000, so that the divisor is a whole number.

1. Multiply both numbers by 1000. (1 mark)
   7.5 ÷ 0.025 = 7500 ÷ 25
2. Divide. 25 goes into 7500 exactly 300 times: 25 × 300 = 7500 (1 mark).
   7500 ÷ 25 = 300
3. State the conclusion. (1 mark)
   So 7.5 ÷ 0.025 = 300.

Check. 300 × 0.025 = 7.5. ✓