Printable · GCSE Foundation · ages 14-16
Number worksheet — GCSE Foundation
Fifteen questions across the number statements at Foundation tier. Choose the non-calculator filter to rehearse Paper 1, which counts for a third of the marks.
Answer key: Number worksheet — GCSE Foundation
MathsUKwww.geekhero.co.uk
- (c) 9/10 — Method: write the decimal over 10 using its one decimal place. Working: 0.9 = 9/10, which is already in its simplest form since 9 and 10 share no common factor. Answer: 9/10. The student's fraction, 9/100, comes from always writing the denominator as 100, regardless of how many decimal places the number actually has. 9 comes from dropping the decimal point altogether and treating 0.9 as the whole number 9. 1/9 comes from flipping the correct fraction upside down.
- (a) 12 — List the multiples of each number: multiples of 4 are 4, 8, 12, 16, 20, 24; multiples of 6 are 6, 12, 18, 24. The lowest number that appears in both lists is 12. Picking 24, a common multiple but not the lowest one, gives an answer that is too big. Picking 6, the larger of the two original numbers rather than a common multiple, ignores that the lowest common multiple must appear in both lists. Working out the highest common factor instead of the lowest common multiple gives 2. So the lowest common multiple of 4 and 6 is 12.
- (a) to the nearest centimetre — Method: the error interval of a rounded measurement runs from half a unit below the stated value to half a unit above it, so the width of the interval is one whole unit of the accuracy used. Working: the interval runs from 24.5 to 25.5, a width of 25.5 − 24.5 = 1, so the unit of accuracy is 1 cm; the stated value is the midpoint, 25 cm, and 25 correct to the nearest centimetre is exactly what gives 24.5 ≤ L < 25.5. Answer: to the nearest centimetre. To the nearest 0.5 cm comes from reading the half-unit, 0.5, as the accuracy itself instead of doubling it back to the full unit. To 1 decimal place comes from seeing the bounds written with one decimal place and taking that as the accuracy, but the bounds of a value given to 1 decimal place would be only 0.05 either side. To the nearest 10 cm comes from confusing the size of the value, about 25, with the unit it was rounded to; rounding to the nearest 10 cm would give an interval 5 cm either side of the stated value.
- (b) 4/9 — Method: double the butter part of the ratio, keeping flour the same, find the new total, then write butter's part over the new total. Working: the new ratio is flour : butter = 5 : 4, since butter doubles from 2 to 4. New total = 5 + 4 = 9. Fraction of butter = 4/9. Answer: 4/9. 2/9 comes from forgetting to double the butter part and using the original value 2 over the new total of 9. 4/7 comes from doubling the butter part correctly to 4 but keeping the old total of 7 instead of working out the new total. 2/5 comes from using the original ratio 5:2 directly as butter over flour without doubling anything.
- (d) 0.3 — Converting the fractions to decimals, 1/4 = 0.25 and 2/5 = 0.4, so any decimal between 0.25 and 0.4 is a valid answer, and 0.3 fits. Confusing 1/4 with 1/5 and converting it as 0.2 instead of 0.25 gives a value below the true lower bound. Confusing 2/5 with 1/2 and converting it as 0.5 instead of 0.4 gives a value above the true upper bound. Converting the fractions correctly but choosing a decimal above the true upper bound of 0.4 instead of between the two values gives 0.45.
- (d) £143 — From 09:15 to 12:45 is 3 hours 30 minutes, which is 3.5 hours. The charge for the time is 3.5 × £28 = £98, and the call-out fee is added on once: £98 + £45 = £143. £98 leaves out the call-out fee altogether, £129 charges for 3 hours instead of 3.5 hours, and £255.50 adds the fee to the hourly rate before multiplying, which charges the £45 for every hour.
- (c) 12 — Method: a square root asks which positive number multiplied by itself gives the number under the root sign. Working: 11 × 11 = 121, which is too small, and 12 × 12 = 144. Answer: 12. The distractors: 14 comes from recalling the wrong square fact, since 14 × 14 = 196; 36 comes from dividing 144 by 4 instead of taking a square root; 72 comes from halving 144, treating a square root as a halving.
- (b) 9/4 — Method: write the whole part as a fraction with the same denominator, then add the fraction part to it. Working: there are 4 quarters in 1 whole, so 2 wholes are 2 × 4 = 8 quarters; adding the 1 quarter that is already there gives 8 + 1 = 9 quarters over a denominator of 4. Answer: 9/4. The distractors: 3/4 comes from adding the whole number to the numerator, as 2 + 1, instead of multiplying it by the denominator first; 7/4 comes from multiplying correctly but then subtracting the numerator, as 2 × 4 − 1; 5/4 comes from multiplying the numerator by the denominator instead of the whole number, as 1 × 4 + 1.
- (a) 3 — Method: the fourth root of a number is the positive value that gives that number when it is multiplied by itself four times. Working: 2 × 2 × 2 × 2 = 16, which is too small, and 3 × 3 × 3 × 3 = 9 × 9 = 81. Answer: 3. The distractors: 9 comes from taking the square root of 81 instead of its fourth root; 4.5 comes from taking the square root and then halving it, as though a fourth root were half a square root; 20.25 comes from dividing 81 by 4, treating the root's index as a divisor.
- (c) 9 — Method: factors come in pairs that multiply to give the number, so work through the pairs in order; a factor paired with itself is counted only once. Working: the pairs are 1 × 100, 2 × 50, 4 × 25, 5 × 20 and 10 × 10. The first four pairs give eight different factors, and the last pair adds only one more, so the factors are 1, 2, 4, 5, 10, 20, 25, 50 and 100. Answer: 9. The distractors: 10 comes from counting the pair 10 × 10 as two separate factors; 8 comes from leaving 1 out of the list, on the view that 1 is not a proper factor; 4 comes from writing 100 = 2² × 5² and multiplying the two indices together instead of adding 1 to each index first.
- (c) 0.0065 — Leading zeros are never significant, so counting from the first non-zero digit, the first two significant figures of 0.006482 are 6 and 4. Look at the next digit along, 8, to decide whether the second figure rounds up: since 8 is 5 or more, the 4 rounds up to 5, giving 0.0065. Rounding to 2 decimal places instead of 2 significant figures gives 0.01, which answers a different question. Wrongly counting one of the leading zeros as a significant figure and stopping one figure short gives 0.006. Keeping an extra digit, as in 0.00648, gives 3 significant figures rather than 2.
- (b) 3 — Method: solving an index equation like this means finding how many factors of the base multiply together to give the number on the right. Working: 4¹ = 4, 4² = 16 and 4³ = 64, so three factors of 4 are needed. Answer: 3. The distractors: 4 comes from listing 4, 16 and 64 and counting the base itself as a step, which gives one more than the index; 6 comes from solving the equation with 2 as the base instead of 4, since 2⁶ = 64; 16 comes from dividing 64 by 4, treating the index as an instruction to divide.
- (a) 360 — The inverse of ÷ 15 is × 15, so the missing number is 24 × 15 = 360. Subtracting instead of multiplying gives 24 − 15 = 9. Dividing by 15 again instead of multiplying gives 24 ÷ 15 = 1.6. Adding instead of multiplying gives 24 + 15 = 39.
- (b) £96.90 — Method: multiply the number of units by the cost per unit, then convert the result from pence to pounds. Working: 340 × 28.5p = 9690p; converting to pounds, 9690p ÷ 100 = £96.90. Answer: £96.90. £969.00 comes from misplacing the decimal point when converting pence to pounds, dividing by 10 instead of 100. £9.69 comes from misplacing the decimal point the other way, dividing by 1000 instead of 100. £102.00 comes from rounding the rate to 30p per unit before multiplying.
- (a) y¹² — Method: when a power is raised to another power, multiply the two indices. Working: (y³)⁴ means y³ × y³ × y³ × y³, which is four lots of three y's multiplied together, so the index is 3 × 4 = 12 and (y³)⁴ = y¹². y⁷ comes from adding the indices, 3 + 4 = 7, which is the rule for multiplying two separate powers, not for raising a power to a power. y⁸¹ comes from working out 3⁴ = 81 and using that as the index, raising the inner index to the outer power instead of multiplying the two indices. 12y comes from multiplying the indices to make 12 but then treating y as a coefficient instead of a power. Answer: y¹².
Build your own mix at the worksheet builder.