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.
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Number worksheet — GCSE Foundation
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- 1.4ˣ = 64. Work out the value of x.
- 2.Given that 2⁵ = 32, work out 2⁶.
- 3.In a photo album the ratio of colour photos to black-and-white photos is 9 : 4. Work out the number of black-and-white photos as a fraction of the number of colour photos.
- 4.Work out 20 − 8 ÷ 2 + 1
- 5.At a book fair the ratio of fiction books to non-fiction books is 4 : 5. Of the non-fiction books, 3/5 are about history. Work out the fraction of all the books at the fair that are history books.
- 6.A train leaves Leeds at 14:35. The journey takes 1 hour 50 minutes. Work out the time the train arrives. Give your answer using the 24-hour clock.
- 7.A fabric is dyed blue and yellow in the ratio 3 : 5. A tailor uses 1.5 m of blue fabric and the matching amount of yellow fabric needed for the ratio. Work out the total length of fabric used.
- 8.Work out ∛(−27) + ∛8
- 9.A space probe is 7.5 × 10⁸ km from Earth. Write this distance as an ordinary number.
- 10.A restaurant offers a lunch deal of one starter from 5 options, one main from 6 options and one dessert from 3 options. Work out how many different lunch deals are possible.
- 11.Two-digit numbers are made using the digits 1, 2, 3 and 4, and no digit may be used twice in the same number. Work out how many different two-digit numbers can be made.
- 12.Simplify (y³)⁴, giving your answer as a single power of y.
- 13.Work out 5 + 3 × (9 − 6)
- 14.A concert hall has 300 seats. 20% of the seats are in the balcony. Work out how many of the seats are in the balcony.
- 15.A block of butter has a mass of 250 g, correct to the nearest 10 g. Using m for the mass of the block in grams, write down the error interval for m.
Answer key
- (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.
- (d) 64 — Method: each time the power increases by 1, the value doubles, since one more 2 is multiplied in. Working: 2⁶ = 2⁵ × 2 = 32 × 2 = 64. Answer: 64. (37 comes from adding the power to the value, 32 + 5, instead of multiplying by the base. 192 comes from multiplying 32 by the new power, 6, instead of by the base, 2. 34 comes from adding the base to the value, 32 + 2, instead of multiplying.)
- (a) 4/9 — Method: write the number of parts for the photos being compared over the number of parts for the photos they are being compared to. Working: black-and-white photos are 4 parts and colour photos are 9 parts, so black-and-white photos as a fraction of colour photos is 4/9. Answer: 4/9. 9/4 comes from writing the ratio the wrong way round, comparing colour to black-and-white instead of black-and-white to colour. 4/13 comes from comparing black-and-white photos to the total number of photos (4+9=13) instead of to the colour photos. 9/13 comes from comparing colour photos to the total number of photos instead of black-and-white to colour.
- (d) 17 — 8 ÷ 2 = 4, then 20 − 4 = 16, then 16 + 1 = 17. Stopping after the subtraction and forgetting to add the final 1 leaves 16. Adding the 4 and the 1 together before subtracting gives 4 + 1 = 5, then 20 − 5 = 15 — the subtraction should use the 4 from the division, not a combined total. Working strictly left to right without giving division priority gives 20 − 8 = 12, then 12 ÷ 2 = 6, then 6 + 1 = 7.
- (c) 1/3 — Method: find non-fiction's fraction of the whole, then multiply by the fraction of non-fiction that is history. Working: total parts = 4 + 5 = 9, so non-fiction is 5/9 of all books. History books are 3/5 of the non-fiction books: 3/5 × 5/9 = 15/45 = 1/3. Answer: 1/3. 3/5 comes from giving the fraction of non-fiction books that are history, without relating it to all the books at the fair. 5/9 comes from stopping after finding the fraction of all books that are non-fiction, without finding the history books within that. 4/15 comes from multiplying 3/5 by the fraction that is fiction (4/9) instead of the fraction that is non-fiction (5/9).
- (c) 16:25 — Add the whole hour first: 14:35 plus 1 hour is 15:35. Then add the 50 minutes. From 15:35, 25 minutes reaches 16:00 and the remaining 25 minutes gives 16:25. 15:85 adds 35 and 50 to get 85 minutes and never exchanges 60 of them for an hour, 16:05 treats 1 hour 50 minutes as 1.5 hours, and 12:45 subtracts the journey time instead of adding it.
- (b) 4 m — Blue fabric is 3 parts and this equals 1.5 m, so one part is 1.5 ÷ 3 = 0.5 m. Yellow fabric is 5 parts, so it is 5 × 0.5 = 2.5 m. The total length is 1.5 + 2.5 = 4 m. 2.5 m is the length of yellow fabric only, without adding the blue fabric back in. 1.5 m is just the given length of blue fabric, with the yellow fabric never worked out. 2.4 m comes from swapping the ratio, treating blue as 5 parts and yellow as 3 parts, giving one part as 1.5 ÷ 5 = 0.3 m and yellow as 3 × 0.3 = 0.9 m, then adding 1.5 + 0.9 = 2.4.
- (c) −1 — Method: find each cube root separately, keeping its sign, and then add the two results. Working: (−3) × (−3) × (−3) = −27, so ∛(−27) = −3, and 2 × 2 × 2 = 8, so ∛8 = 2. Adding gives −3 + 2 = −1. Answer: −1. The distractors: 5 comes from taking the cube root of a negative number as positive, giving 3 + 2; −5 comes from reading the minus sign as applying to the whole sum and working out −(3 + 2); −6 comes from multiplying the two roots, −3 × 2, instead of adding them.
- (b) 750,000,000 — Method: multiplying by 10⁸ moves the decimal point eight places to the right, and every empty place is filled with a zero. Working: 10⁸ = 100,000,000, and moving the decimal point in 7.5 eight places to the right gives 7.5 × 100,000,000. Answer: 750,000,000. The distractors: 7,500,000,000 comes from removing the decimal point first to make 75 and then writing eight zeros after it, which carries the digits one place too far; 600 comes from reading 10⁸ as 10 × 8 = 80 and working out 7.5 × 80; 0.000000075 comes from moving the decimal point eight places to the left, as though the index were negative.
- (c) 90 — Multiply the number of choices for each course: 5 × 6 × 3 = 90. 14 comes from adding the three numbers instead of multiplying them. 30 comes from multiplying only the starters and mains, 5 × 6, and forgetting the dessert. 18 comes from multiplying only the mains and desserts, 6 × 3, and forgetting the starter.
- (a) 12 — Method: build the number one place at a time, listing systematically: fix the tens digit, then run through every units digit that is still available. Working: any of the 4 digits can go in the tens place, and once it has been used only 3 digits are left for the units place, so there are 4 × 3 = 12 numbers; listing the numbers that begin with 1 gives 12, 13 and 14, and each of the other three starting digits gives 3 numbers in the same way. Answer: 12. The distractors: 16 comes from working out 4 × 4, which allows a digit to be used twice; 8 comes from multiplying the 4 digits by the 2 places in the number instead of multiplying the choices available at each place; 6 comes from treating a number and its reverse as the same, counting only the unordered pairs of digits.
- (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.
- (a) 14 — Brackets first: 9 − 6 = 3. Then multiply: 3 × 3 = 9. Then add: 5 + 9 = 14. So the answer is 14. A candidate who worked out (5 + 3) × (9 − 6) = 8 × 3 = 24 added before multiplying, ignoring the priority of operations outside the bracket. A candidate who dropped the brackets and worked out 5 + 3 × 9 − 6 = 5 + 27 − 6 = 26 multiplied by the 9 itself instead of by the bracket's value of 3, losing the grouping the brackets give. A candidate who forgot to add the 5 and only worked out 3 × (9 − 6) = 3 × 3 = 9 dropped a term from the calculation.
- (d) 60 — Method: a percentage acts as an operator, so finding 20% of an amount means multiplying it by 20/100, which cancels to 1/5. Working: 20% = 20/100 = 1/5, and 300 ÷ 5 = 60. Answer: 60 seats. The distractors: 15 comes from reading 20% as one twentieth and working out 300 ÷ 20 = 15; 30 comes from finding 10% of 300 and stopping there instead of doubling it; 6 comes from converting 20% to 0.02 rather than 0.2, giving 0.02 × 300 = 6.
- (b) 245 ≤ m < 255 — Method: a mass given to the nearest 10 g lies within half of 10 g, that is 5 g, of the figure written down. The lower limit is included because it rounds up to that figure, and the upper limit is excluded because it rounds up to the next multiple of 10. Working: 250 − 5 = 245 and 250 + 5 = 255, and a mass of 245 g rounds to 250 g while a mass of 255 g rounds to 260 g. Answer: 245 ≤ m < 255. The distractors: 240 ≤ m < 260 goes a whole 10 g either side instead of half of it; 245 < m ≤ 255 has the two limits the wrong way round; 240 ≤ m < 250 treats the stated 250 g as the largest mass possible, as though rounding were always upwards.
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