Number

In the number 4,206.57, what digit is in the hundredths place?

• A0
• B7✓
• C5
• D6

Answer: The hundredths place is the second digit after the decimal point. In 4,206.57 that digit is 7, since the 5 sits in the tenths place. The 6 is the units digit and the 0 is the tens digit, so both belong to the whole-number part.

Which of these four numbers is the largest: 0.4500, 0.500, 0.39800, 0.41?

• A0.41
• B0.39800
• C0.45
• D0.500✓

Answer: Comparing decimals means lining up the place values, not counting digits. 0.500 is bigger than 0.4500, 0.39800 and 0.41 because its tenths digit, 5, is the largest of the four. Trailing zeros do not change a decimal's value, so 0.4500 is still less than 0.500, and a common slip is assuming a longer string of digits makes a number bigger.

In a quiz, four teams finish with these point totals: -3, 145, 156 and 108. Which team has the lowest score?

• A145 points
• B156 points
• C-3 points✓
• D108 pts

Answer: The lowest score is the smallest number on the number line, and -3 is less than every positive score, so the team with -3 points has the lowest score. All three of the positive totals, 145, 156 and 108, are greater than -3, since any positive number is greater than any negative number.

Which of these fractions has the smallest value: 7/12, 11/15, 13/20, 3/4?

• A13/20
• B3/4
• C11/15
• D7/12✓

Answer: Converting each fraction to sixtieths gives 35/60, 44/60, 39/60 and 45/60, so 7/12 is the smallest. A common mistake is assuming the fraction with the largest denominator must be smallest, which is wrong here since 11/15 and 13/20 both have larger denominators than 7/12 but represent bigger values.

Which of these numbers is prime: 51, 57, 61, 63?

• A57
• B63
• C51
• D61✓

Answer: A prime number has exactly two factors, itself and 1. 61 cannot be divided evenly by any number except 1 and 61. The other three all factorise: 51 is 3 times 17, 57 is 3 times 19, and 63 is 7 times 9.

A gardener has 36 tulip bulbs and 48 daffodil bulbs and wants to split them into the largest possible number of identical bags with no bulbs left over. What is the highest common factor of 36 and 48?

• A24
• B12✓
• C36
• D18

Answer: Writing 36 as 2 squared times 3 squared and 48 as 2 to the power 4 times 3 shows the shared factors are 2 squared and 3, giving 12. Twenty-four divides 48 but not 36, eighteen divides 36 but not 48, and 36 is simply one of the original numbers rather than a factor common to both.

Two bus routes leave the depot at the same time. One leaves every 8 minutes and the other every 12 minutes. After how many minutes will they next leave together?

• A48
• B96
• C12
• D24✓

Answer: The buses next leave together after the lowest common multiple of 8 and 12, which is 24. Ninety-six comes from multiplying 8 by 12 without removing their shared factor of 4, 48 is a common multiple but not the lowest one, and 12 is simply one of the two original numbers.

Written as a product of its prime factors, what is 60?

• A2 cubed times 3 times 5
• B2 squared times 3 times 5✓
• C3 squared times 2 times 5 times 1
• D2 times 3 times 10

Answer: Repeatedly dividing 60 by prime numbers gives 60 = 2 times 2 times 3 times 5, which is 2 squared times 3 times 5. Squaring the 3 instead of the 2 gives 3 squared times 2 times 5 times 1, which is actually 90, cubing the 2 instead of squaring it gives 2 cubed times 3 times 5 times 1, which is actually 120, and stopping the factorising too early leaves the non-prime factor 10 in 2 times 3 times 10.

Given that 90 = 2 times 3 squared times 5 and 126 = 2 times 3 squared times 7, what is the highest common factor of 90 and 126?

• A90
• B18✓
• C27
• D15

Answer: The highest common factor uses only the prime factors that appear in both lists, taken to the lower power, which here is 2 times 3 squared, or 18. Using 3 cubed instead of 2 times 3 squared gives 27, mistaking one of the original numbers for their highest common factor gives 90, and leaving out one of the factors of 3 gives 15.

Given that 24 = 2 cubed times 3 and 90 = 2 times 3 squared times 5, what is the lowest common multiple of 24 and 90?

• A1080
• B2160
• C360✓
• D12

Answer: The lowest common multiple takes each prime factor to the higher power seen in either list, giving 2 cubed times 3 squared times 5, which equals 360. Multiplying 24 by 90 directly gives 2160, forgetting to divide out a shared factor of 2 gives 1080, and mistaking half of 24 for the lowest common multiple gives just 12.

A submarine is at a depth of -140 m relative to the surface. It rises 55 m. What is its new depth?

• A-95 m
• B85 m
• C-195 m
• D-85 m✓

Answer: Rising reduces the size of the negative depth, so -140 plus 55 gives -85 m, still below the surface. Dropping the negative sign gives 85 m, adding the rise the wrong way round gives -195 m, and a subtraction slip when combining 140 and 55 gives -95 m.

Work out -68 minus (-45).

• A113
• B23
• C-23✓
• D-113

Answer: Subtracting a negative number is the same as adding its positive value, so -68 minus (-45) becomes -68 plus 45, which is -23. Treating both signs as negative and adding the magnitudes gives -113, dropping the negative sign gives 23, and adding the two magnitudes without applying the correct sign gives 113.

Work out (-12) times (-13).

• A-25
• B-156
• C156✓
• D25

Answer: A negative number multiplied by a negative number always gives a positive result, so (-12) times (-13) equals 156, since 12 times 13 is 156. Forgetting that rule and keeping the answer negative gives -156, while adding the two numbers instead of multiplying them gives 25, or -25 if the negative sign is mistakenly kept as well.

Work out -92 divided by 4.

• A-23✓
• B23
• C-96
• D-368

Answer: A negative number divided by a positive number gives a negative answer, and 92 divided by 4 is 23, so the result is -23. Dropping the negative sign gives 23, subtracting 4 from -92 instead of dividing gives -96, and multiplying -92 by 4 instead of dividing gives -368.

Work out 3/4 plus 5/6, giving your answer in its simplest form.

• A19/24
• B1 7/12✓
• C17/12
• D8/10

Answer: Converting to twelfths gives 9/12 plus 10/12, which is 19/12, or 1 and 7/12 as a mixed number. Adding the numerators and denominators straight across gives 8/10, using 24 as the common denominator without simplifying gives 19/24, and mishandling the improper fraction's whole-number part gives 1 and 17/12.

A recipe needs 3 and 3/4 cups of flour, but 1 and 1/6 cups have already been added. How many more cups are needed?

• A2 7/12✓
• B2 5/12
• C7/12
• D2 8/12

Answer: Converting to twelfths gives 3 and 9/12 minus 1 and 2/12, which leaves 2 and 7/12. Writing 1/6 as 4/12 instead of 2/12 gives 2 and 5/12, writing it as 1/12 gives 2 and 8/12, and forgetting to subtract the whole numbers gives just 7/12.

Work out 3/5 times 2/9, giving your answer in its simplest form.

• A27/45
• B2/15✓
• C1/3
• D4/5

Answer: Multiplying the numerators and denominators gives 6/45, which simplifies to 2/15 once both are divided by 3. Cross-multiplying the wrong pair of numbers gives 27/45, doubling the numerator by mistake gives 12/45, and cancelling only part of the fraction gives 1/3.

A ribbon 5/6 of a metre long is cut into pieces that are each 1/12 of a metre long. How many pieces are made?

• A1/10 pieces
• B5/72 pieces
• C120 pcs
• D10 pieces✓

Answer: Dividing by a fraction means multiplying by its reciprocal, so 5/6 divided by 1/12 becomes 5/6 times 12, which is 10 pieces. Multiplying the two fractions directly without flipping the second one gives 5/72 pieces, flipping the wrong fraction gives 1/10 pieces, and muddling the twelfths with the ribbon's total length gives 120 pieces.

A winter coat costs £145.60 and a pair of boots costs £12.85. What is the total cost of both items?

• A£58.45
• B£1,584.50
• C£157.45
• D£158.45✓

Answer: Adding the pounds and pence separately gives £145.60 plus £12.85, which totals £158.45. Misplacing the decimal point when adding gives £1,584.50, forgetting the hundreds digit of the coat's price gives £58.45, and a carrying slip in the pence column gives £157.45.

Work out 3.6 times 0.8.

• A4.4
• B28.8
• C2.88✓
• D0.288

Answer: Multiplying 36 by 8 gives 288, and since the two original numbers together have two decimal places, the answer is 2.88. Placing the decimal point one place too far right gives 28.8, placing it one place too far left gives 0.288, and adding the numbers instead of multiplying gives 4.4.

Work out 8 plus 4 times (6 minus 2), applying the correct order of operations.

• A48
• B16
• C30
• D24✓

Answer: The bracket is evaluated first, giving 6 minus 2 equals 4, then the multiplication 4 times 4 equals 16, and finally 8 is added to give 24. Adding 8 and 4 before multiplying by the bracket gives 48, ignoring the brackets entirely gives 30, and forgetting to add the 8 at the end gives 16.

Work out (2 plus 3) squared minus 4 times 2.

• A14
• B21
• C17✓
• D42

Answer: The bracket gives 5, and 5 squared is 25, so the expression becomes 25 minus 8, which is 17. Subtracting before multiplying gives 42, forgetting to multiply the 4 by 2 before subtracting gives 21, and squaring only the 3 inside the bracket before continuing gives 14.

A number is multiplied by 4 and then 7 is added, giving a result of 79. What was the original number?

• A20
• B18✓
• C72
• D86

Answer: Using inverse operations in reverse order, 7 is subtracted from 79 to give 72, and then 72 is divided by 4 to give 18. Stopping after the subtraction without dividing gives 72, adding 7 instead of subtracting it gives 86, and dividing 79 by 4 first without removing the 7 gives roughly 20.

What is the value of the square root of 144?

• A72
• B11
• C12✓
• D14

Answer: The square root of 144 is the number that multiplies by itself to give 144, and 12 times 12 equals 144. Eleven is close but 11 times 11 is only 121, halving 144 gives 72, and fourteen times fourteen overshoots to 196.

A cube-shaped gift box has a volume of 125 cubic cm. What is the length of each edge?

• A42 cm
• B15 cm
• C5 cm✓
• D25 cm

Answer: The edge length of a cube is the cube root of its volume, and 5 times 5 times 5 equals 125, so each edge is 5 cm. Squaring 5 instead of cubing it suggests 25 cm, dividing 125 by 3 instead of taking the cube root gives roughly 42 cm, and multiplying 5 by 3 instead of cubing it gives 15 cm.

What is the value of 2 to the power 7?

• A49
• B14
• C128✓
• D64

Answer: 2 to the power 7 means seven 2s multiplied together, which is 2 times 2 times 2 times 2 times 2 times 2 times 2, giving 128. Multiplying the base by the power instead of repeating it gives 14, swapping the base and the power to get 7 squared gives 49, and stopping one power short (2 to the power 6) gives 64.

The square root of 98 lies between which two consecutive whole numbers?

• A10 and 11
• B7 and 8
• C8 and 9
• D9 and 10✓

Answer: Since 9 squared is 81 and 10 squared is 100, and 98 sits between 81 and 100, the square root of 98 must lie between 9 and 10. Using 8 squared and 9 squared, which are 64 and 81, wrongly suggests 8 and 9, using 10 squared and 11 squared wrongly suggests 10 and 11, and using 7 squared and 8 squared, which are 49 and 64, wrongly suggests 7 and 8.

A stadium has 58,000 seats. Write 58,000 in standard form.

• A58 times 10 to the power 3
• B5.8 times 10 to the power 4✓
• C5.8 times 10 to the power 3
• D5.8 times 10 to the power 5

Answer: Standard form needs a number between 1 and 10 multiplied by a power of 10, and moving the decimal point four places from 58000 gives 5.8 times 10 to the power 4. Using one power too many gives 5.8 times 10 to the power 5, leaving the coefficient as 58 breaks the between-1-and-10 rule, and using one power too few gives 5.8 times 10 to the power 3.

A company's turnover is 3.2 times 10 to the power 6 pounds. Write this as an ordinary number.

• A£3,200
• B£3,200,000✓
• C£320,000
• D£3,200,000,000

Answer: Multiplying by 10 to the power 6 means moving the decimal point six places to the right, turning 3.2 into 3,200,000. Moving the point only three places gives £3,200, moving it only five places gives £320,000, and moving it nine places (three too many) gives £3,200,000,000.

Write 0.35 as a fraction in its simplest form.

• A7/200
• B7/10
• C35/10
• D7/20✓

Answer: 0.35 means 35 hundredths, so it is written as 35/100, which simplifies to 7/20 by dividing top and bottom by 5. Treating the digits as tenths instead of hundredths gives 35/10, dividing only the numerator by 5 gives 7/10, and shifting the decimal point the wrong way when converting gives 7/200.

What is 15% of £84?

• A£8.40
• B£12.60✓
• C£1.26
• D£126.00

Answer: Finding 15% means multiplying £84 by 0.15, which gives £12.60. Finding 10% instead gives £8.40, misreading 15% as 150% gives £126.00, and misplacing the decimal point gives £1.26.

A rucksack's price rises from £40 to £46. What is the percentage increase?

• A15%✓
• B115%
• C6%
• D13%

Answer: The increase of £6 is compared to the original price of £40, and 6 divided by 40 as a percentage is 15%. Comparing the increase to the new price of £46 instead gives roughly 13%, comparing the new price to the original without subtracting 100% gives 115%, and stating the raw increase in pounds as if it were the percentage gives 6%.

Last year a chess club had 60 members. This year it has 90 members. Express this year's membership as a percentage of last year's.

• A50%
• B150%✓
• C66.7%
• D33%

Answer: This year's total is compared to last year's total, so 90 divided by 60 as a percentage is 150%, showing the club is now half as big again. Reporting only the percentage increase rather than the whole percentage gives 50%, dividing the numbers the wrong way round gives roughly 66.7%, and comparing the increase to the new total instead of the original gives roughly 33.3%.

Round 5.987 to 1 decimal place.

• A5.98
• B6.0✓
• C5.99
• D6.1

Answer: Rounding to 1 decimal place means looking at the second decimal digit, 8, which is 5 or more and so rounds the first decimal digit up, turning 5.987 into 6.0. Truncating instead of rounding gives 5.98, rounding to 2 decimal places instead of 1 gives 5.99, and rounding the first decimal digit the wrong way gives 6.1.

A plank's length is measured as 14 cm, correct to the nearest centimetre. What is the error interval for the true length, l?

• A13.5 cm is less than or equal to l, which is less than 14.5 cm✓
• B13 cm is less than or equal to the true length l, which is less than 15 cm
• C13.5 cm is strictly less than the true length l, which is strictly less than 14.5 cm
• D14 cm is less than or equal to l, which is less than 14.5 cm

Answer: Rounding to the nearest centimetre means the true length can be up to half a centimetre smaller or larger than 14 cm, giving the interval from 13.5 cm up to but not including 14.5 cm. Using a whole centimetre either side instead of half gives too wide a range, only allowing the length to round up misses the lower half of the interval, and excluding the lower bound with a strict inequality ignores that a length of exactly 13.5 cm also rounds to 14 cm.