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.
Number worksheet — GCSE Foundation
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- 1.Which of these numbers lies between −8 and −5 on a number line?
- 2.A crowd of 8,400 people is recorded correct to the nearest 100. Work out the smallest possible number of people in the crowd.
- 3.A laptop costs £720. A carrying case for it costs 1/9 of the price of the laptop. Work out the cost of the carrying case.
- 4.A photograph uses 4 × 10⁶ bytes of storage. A memory card holds 3.2 × 10¹⁰ bytes. Work out how many of these photographs the card can hold. Give your answer in standard form.
- 5.A quiz has 50 questions. Freya answers 10% of them incorrectly. Work out how many questions she answers incorrectly.
- 6.A jar of jam costs £2.35. Tom buys 3 jars. He says the total cost is more than £7. Work out the exact total cost, and state whether Tom's claim is correct.
- 7.A grain of fine sand has a mass of 0.001 grams. Write this mass in standard form.
- 8.Work out (2 × 10³) × (3 × 10⁴). Give your answer in standard form.
- 9.Write 0.25 as a fraction in its simplest form.
- 10.A student always writes the digits after the decimal point over a denominator of 100, whatever the number of decimal places. Using this wrong rule, they convert 0.9 to a fraction. Work out the correct fraction that 0.9 is equal to, giving your answer in its simplest form.
- 11.The number of pages in a book is 240, correct to the nearest 10 pages. Write down the error interval for the actual number of pages, n.
- 12.£60 is shared between Amir, Bo and Chen in the ratio 1:2:3. Work out the fraction of the £60 that Amir and Bo receive together.
- 13.Put these numbers in order, starting with the smallest: 3.2 × 10⁴, 2.9 × 10⁵, 4.1 × 10³
- 14.Round 6.283 to 1 significant figure.
- 15.Work out 2/3 × 3/4 exactly, giving your answer in its simplest form.
Answer key
- (d) −6 — Method: check each option to see if it falls strictly between −8 and −5, remembering that numbers become smaller as they get more negative. Working: −8 < −6 < −5, so −6 lies between them. Answer: −6. −9 comes from a value that is more negative than −8, so it lies outside the range, below −8. −4 comes from a value that is less negative than −5, so it lies outside the range, above −5. 6 comes from ignoring the negative signs and picking a positive number between 5 and 8.
- (a) 8,350 — Rounding to the nearest 100 means the true number can be up to half of 100, which is 50, below the recorded figure before it would round down to a lower hundred. The smallest possible number is therefore 8,400 − 50 = 8,350. Adding 50 instead of subtracting it gives 8,450, which is the upper end of the interval rather than the smallest value, and 8,450 is not itself possible because it would round up to 8,500. Subtracting a whole 100 instead of half of it gives 8,300, going too far below the recorded value. Subtracting 10 instead of half of the rounding unit gives 8,390, treating the rounding unit as 100 but the tolerance as only 10.
- (c) £80 — Method: a unit fraction acts as an operator, so finding 1/9 of a price means dividing that price by 9. Working: £720 ÷ 9 = £80. Answer: £80. The distractors: £6480 comes from multiplying by the denominator instead of dividing by it, giving £720 × 9 = £6480; £640 comes from working out what is left of the £720 once the case is paid for, £720 − £80, instead of the cost of the case itself; £72 comes from dividing by 10 instead of 9, treating one ninth as one tenth.
- (a) 8 × 10³ — Method: divide the capacity of the card by the size of one photograph, dividing the coefficients and subtracting the indices, then bring the coefficient back into the range 1 to 10. Working: 3.2 ÷ 4 = 0.8 and 10 − 6 = 4, which gives 0.8 × 10⁴; a coefficient of 0.8 is smaller than 1, so the decimal point moves one place to the right and the index falls by 1. Answer: 8 × 10³. The distractors: 8 × 10⁴ comes from correcting 0.8 to 8 without reducing the index, which makes the answer ten times too large; 1.28 × 10¹⁷ comes from multiplying the two numbers instead of dividing them, since 3.2 × 4 = 12.8 and 10 + 6 = 16; 8 × 10¹⁵ comes from dividing the coefficients but adding the indices instead of subtracting them.
- (a) 5 — Method: 10% is 10/100, which cancels to 1/10, so finding 10% of an amount means dividing it by 10. Working: 10% = 10/100 = 1/10, and 50 ÷ 10 = 5. Answer: 5 questions. The distractors: 10 comes from reading the 10% as a count of 10 questions rather than as an operator acting on the 50; 45 comes from working out how many questions Freya answers correctly, the other 90% of the quiz, instead of how many she gets wrong; 500 comes from multiplying by 10 instead of dividing by 10.
- (c) £7.05 — Tom is correct — Method: multiply the cost of one jar by the number of jars, then compare the total to £7. Working: 3 × £2.35 = £7.05, and since £7.05 is more than £7, Tom's claim is correct. Answer: £7.05 — Tom is correct. "£7.05 — Tom is incorrect" comes from reaching the right total but reading the 05 after the point as making the amount less than £7. "£6.95 — Tom is incorrect" comes from working out 3 × 35p as 3 × 30p + 5p = 95p, multiplying only the tens digit, and adding it to 3 × £2 = £6. "£7.20 — Tom is correct" comes from rounding £2.35 up to £2.40 before multiplying: 3 × £2.40 = £7.20.
- (d) 1 × 10⁻³ — Method: for a number smaller than 1 the index is negative, and it counts the places the decimal point moves to the right to leave a coefficient between 1 and 10. Working: the only significant digit is 1, so the coefficient is 1; the decimal point in 0.001 moves three places to the right to reach that 1, so the index is −3. Answer: 1 × 10⁻³. The distractors: 1 × 10³ comes from taking the index as positive, which describes one thousand grams rather than one thousandth of a gram; 0.1 × 10⁻² is the same mass written with a coefficient of 0.1, which is smaller than 1 and so is not standard form; 1 × 10⁻⁴ comes from counting the zero in front of the decimal point as well as the places after it.
- (d) 6 × 10⁷ — Method: the coefficients and the powers of ten are handled separately — multiply the coefficients, and add the indices because the powers share the base 10. Working: 2 × 3 = 6 for the coefficients, and 10³ × 10⁴ = 10⁷ for the powers; 6 lies between 1 and 10, so the coefficient needs no adjustment. Answer: 6 × 10⁷. The distractors: 5 × 10⁷ comes from adding the coefficients, 2 + 3, instead of multiplying them; 6 × 10¹² comes from multiplying the indices, 3 × 4, instead of adding them; 6 × 10¹ comes from subtracting the indices, 4 − 3, which is the rule for dividing rather than for multiplying.
- (d) 1/4 — Method: count the digits after the decimal point to choose a denominator that is a power of ten, then cancel the fraction by the highest common factor of the numerator and the denominator. Working: 0.25 has two digits after the point, so 0.25 is 25 hundredths and can be written as 25/100; the highest common factor of 25 and 100 is 25, and 25 ÷ 25 = 1 with 100 ÷ 25 = 4. Answer: 1/4. The distractors: 2/5 comes from reading the two digits after the point as a numerator and a denominator and writing 2 over 5; 1/25 comes from putting 1 over the digits after the point, as though 0.25 meant one twenty-fifth; 1/40 comes from miscounting the decimal places and using a denominator of 1000, giving 25/1000, which cancels by 25 to 1/40.
- (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.
- (b) 235 ≤ n < 245 — Method: the error interval stretches half the rounding unit either side of the rounded value, with the upper bound excluded because it would round up to the next value. Working: half of 10 is 5, so the interval runs from 240 − 5 to 240 + 5. Answer: 235 ≤ n < 245. (230 ≤ n < 250 comes from using the whole rounding unit, 10, either side instead of half of it. 235 ≤ n ≤ 245 comes from including the upper bound with ≤ instead of excluding it with <. 239.5 ≤ n < 240.5 comes from rounding to the nearest whole number instead of the nearest 10, so half of 1 is used in place of half of 10.)
- (a) 1/2 — The ratio 1:2:3 has 1 + 2 + 3 = 6 parts in total. Amir and Bo together receive 1 + 2 = 3 of those parts, so together they receive 3/6 of the £60, which simplifies to 1/2. Using only Amir's single part, 1/6, ignores Bo's share entirely. Adding Bo's and Chen's parts instead of Amir's and Bo's, 2 + 3 = 5, gives 5/6. Comparing Amir and Bo's 3 parts to Chen's 3 parts, rather than to the total of 6 parts, gives 3/3 = 1.
- (c) 4.1 × 10³, 3.2 × 10⁴, 2.9 × 10⁵ — The exponent decides the size first: 10³ is smaller than 10⁴, which is smaller than 10⁵, so the order is 4.1 × 10³, then 3.2 × 10⁴, then 2.9 × 10⁵. Reversing the whole list gives largest to smallest instead of smallest to largest. Comparing 3.2 × 10⁴ and 4.1 × 10³ by their coefficients alone, 3.2 against 4.1, and swapping them ignores that 10⁴ is bigger than 10³ regardless of the coefficient. Comparing 2.9 × 10⁵ and 3.2 × 10⁴ by their coefficients alone and swapping them makes the same mistake at the top of the list.
- (c) 6 — Method: the first significant figure is the first non-zero digit; round using the digit after it to decide whether to round up or down. Working: the first significant figure of 6.283 is the 6; the next digit is 2, which rounds down, so 6.283 rounds to 6. 6.3 comes from rounding to 2 significant figures instead of 1. 10 comes from rounding up to the nearest 10 instead of finding 1 significant figure of the number itself. 0.6 comes from misplacing the decimal point after rounding. Answer: 6.
- (b) 1/2 — To multiply fractions, multiply the numerators together and multiply the denominators together: 2 × 3 = 6 and 3 × 4 = 12, giving 6/12, which simplifies to 1/2. Adding the fractions instead of multiplying them, using a common denominator of 12, gives 8/12 + 9/12 = 17/12. Dividing by 3/4 instead of multiplying by it, so multiplying by its reciprocal 4/3, gives 2/3 × 4/3 = 8/9. Multiplying only the numerators, 2 × 3 = 6, and keeping the first denominator, 3, unchanged gives 6/3 = 2.
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