Printable · GCSE Higher · ages 14-16
Exact calculation: fractions, surds and π worksheet — GCSE Higher
Fifteen questions on "exact calculation: fractions, surds and π" — DfE statement N8. Print it, or print three versions so neighbours cannot copy by letter; the key gives the letter for each version.
part Higher
Exact calculation: fractions, surds and π worksheet — GCSE Higher
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- 1.A cake recipe needs 3/4 of a kilogram of sugar. Aisha wants to make half the recipe. Work out how much sugar she needs, giving your answer as a fraction of a kilogram in its simplest form.
- 2.The two shorter sides of a right-angled triangle are √12 cm and √24 cm. Work out the exact length of the hypotenuse.
- 3.A square tile has an area of 72 cm². Work out the exact perimeter of the tile, giving your answer in the form k√2 cm.
- 4.Chloe wants to work out 3 1/4 − 1 2/3. She converts both mixed numbers to twelfths, then subtracts the whole numbers and the fraction parts separately, without checking whether she needs to exchange first. Work out the correct value of 3 1/4 − 1 2/3, giving your answer as a mixed number in its simplest form.
- 5.A circle has a radius of 3 cm. Which of these is the exact area of the circle?
- 6.Work out 5/6 × 2/9, giving your answer in its simplest form.
- 7.A quarter-circle has a radius of 6 cm. Work out the exact perimeter of the quarter-circle, giving your answer in terms of π.
- 8.Work out √3 × √12, giving your answer as an integer.
- 9.In a school, 3/5 of the students study French. Of these students, 2/3 also study Spanish. What fraction of all the students study both French and Spanish?
- 10.Expand and simplify √3(2 + √12).
- 11.Rationalise the denominator of 6/√3, giving your answer in its simplest form.
- 12.A circle has a circumference of 8π cm. Work out the exact area of the circle in terms of π.
- 13.Work out 1 1/2 ÷ 3/4 exactly, giving your answer in its simplest form.
- 14.A student works out the exact area of a circle with radius 4 cm by squaring the radius but forgetting to multiply by π. Work out the correct exact area of the circle, in terms of π.
- 15.Work out 3/4 − 5/12 exactly, giving your answer in its simplest form.
Answer key
- (d) 3/8 — Method: making half the recipe means dividing the quantity of sugar by 2. Working: 3/4 ÷ 2 = 3/8. Answer: 3/8. 3/2 comes from multiplying by 2 instead of dividing, as if doubling the recipe. 5/4 comes from adding 1/2 to 3/4 instead of halving it, confusing "half of" with "plus a half". 3/4 comes from leaving the amount unchanged, forgetting to halve it for the smaller recipe.
- (d) 6 — By Pythagoras' theorem, the square of the hypotenuse equals the sum of the squares of the other two sides: (√12)² + (√24)² = 12 + 24 = 36. The hypotenuse is √36 = 6 cm. Adding the two side lengths directly instead of squaring them first, treating the theorem as if it were a straight sum of the sides, gives √12 + √24 = 2√3 + 2√6. Multiplying the two squared values, 12 × 24 = 288, instead of adding them, then taking the root, gives √288 = 12√2. Adding the squares correctly to get 36 but forgetting to take the square root at the end leaves 36 as the answer instead of the hypotenuse itself.
- (b) 24√2 — The side length of the tile is √72. Since 72 = 36 × 2, √72 = √36 × √2 = 6√2 cm. A square has four equal sides, so the perimeter is 4 × 6√2 = 24√2 cm. Simplifying √72 by writing the perfect-square factor itself as the coefficient instead of its root, 36√2 instead of 6√2, and then multiplying by 4 lands on 144√2. Working out the correct side length, 6√2 cm, but then giving that as the final answer without multiplying by 4 for the perimeter gives 6√2. Doubling the side length instead of quadrupling it, as if the perimeter were 2 × 6√2 rather than 4 × 6√2, gives 12√2.
- (d) 1 7/12 — Method: convert both mixed numbers to improper fractions with a common denominator, then subtract. Working: 3 1/4 = 39/12 and 1 2/3 = 20/12, so 39/12 − 20/12 = 19/12 = 1 7/12. Answer: 1 7/12. Chloe's method, subtracting whole numbers (3−1=2) and fraction parts (2/3−1/4=5/12) separately without exchanging, gives 2 5/12. 1 3/4 comes from converting 2/3 to twelfths incorrectly as 6/12 instead of 8/12, then subtracting. 8 comes from converting both mixed numbers to improper fractions correctly (13/4 and 5/3) but then subtracting numerators and denominators separately: (13−5)/(4−3) = 8/1.
- (c) 9π cm² — The area of a circle is π × r². With a radius of 3 cm this is π × 3² = 9π cm², and this is exact because π has not been replaced by any approximation. Writing 28.3 cm² replaces π with a rounded decimal value, 3.14, and then rounds the result again, so it is only an approximation. Writing 28.26 cm² uses π ≈ 3.14 without a final rounding step, but this is still only an approximation of 9π, not the exact value. Writing 27 cm² comes from replacing π with the rough approximation 3, which is even further from the true value.
- (b) 5/27 — Method: multiply the numerators together and the denominators together, then simplify. Working: (5 × 2)/(6 × 9) = 10/54 = 5/27. Answer: 5/27. 7/15 comes from adding the fractions instead of multiplying: (5+2)/(6+9) = 7/15. 15/4 comes from flipping the second fraction, as if dividing: (5 × 9)/(6 × 2) = 45/12 = 15/4. 5/3 comes from cancelling the two denominators against each other, dividing both 6 and 9 by 3 to leave 5/2 × 2/3 = 10/6 = 5/3; cancelling is only valid between a numerator and a denominator, never between two denominators.
- (c) (12 + 3π) cm — The perimeter of a quarter-circle is made up of two straight radii plus a quarter of the circumference. The two radii give 2 × 6 = 12 cm, and a quarter of the circumference is (1/4) × 2 × π × 6 = 3π cm, so the total perimeter is (12 + 3π) cm. Giving only the curved part, 3π cm, forgets the two straight edges entirely. Using the full circumference, 2 × π × 6 = 12π, instead of a quarter of it gives (12 + 12π) cm. Including only one radius instead of two gives (6 + 3π) cm.
- (a) 6 — Use √a × √b = √(ab): √3 × √12 = √(3 × 12) = √36 = 6. Adding the numbers under the roots instead of multiplying them, 3 + 12 = 15, gives √15 — that comes from applying the rule for adding surds to a multiplication question. Multiplying the two numbers under the roots but then forgetting to take the square root at the end leaves 36. Simplifying only √12 to 2√3 and then dropping the other √3 factor entirely gives 2√3.
- (c) 2/5 — To find a fraction of a fraction, multiply them together: 3/5 × 2/3 = 6/15, which simplifies to 2/5. Multiplying only the numerators, 3 × 2 = 6, but adding the denominators, 5 + 3 = 8, instead of multiplying them gives 6/8, which simplifies to 3/4. Using only the fraction who study French, 3/5, and ignoring that a further fraction of them also study Spanish gives 3/5. Dividing by 2/3 instead of multiplying by it, using its reciprocal 3/2, gives 3/5 × 3/2 = 9/10.
- (d) 6 + 2√3 — Multiply √3 by each term in the bracket separately. First term: √3 × 2 = 2√3. Second term: √3 × √12 = √(3 × 12) = √36 = 6. Adding the two results in the order they were found, and writing the whole-number term first, gives 6 + 2√3. Adding the numbers under the root for the second term instead of multiplying them (3 + 12 = 15) gives √15 in place of 6, leading to √15 + 2√3. Multiplying √3 by the 2 but never distributing to the √12 term at all leaves just 2√3. Treating √3 × 2 as if the 3 were multiplied by the 2 inside the root, √3 × 2 → √6, while still getting the second term correct, gives 6 + √6.
- (a) 2√3 — Multiply the top and bottom of the fraction by √3, since √3 × √3 = 3: 6/√3 = (6 × √3)/(√3 × √3) = 6√3/3. Dividing 6 by 3 gives 2, so the fraction simplifies to 2√3. Multiplying only the numerator by √3 and then cancelling the surd in the denominator against it as if they were the same term, without properly squaring the denominator, leads to 6. Dividing 6 by 3 as 3 instead of 2 gives 3√3 — a slip in the final division. Simplifying 6√3/3 by cancelling the whole numerator's 3 with the denominator's 3, including the surd, gives 2, which loses the surd altogether.
- (b) 16π — Use the circumference formula C = 2πr to find the radius: 8π = 2πr, so r = 8π ÷ 2π = 4 cm. Then use the area formula A = πr²: A = π × 4² = 16π cm². Using C = πr instead of C = 2πr gives r = 8, and squaring that gives 64π. Finding r = 4 correctly but then substituting it back into the circumference formula instead of the area formula gives 2π × 4 = 8π. Finding r = 4 correctly but forgetting to square it in the area formula, using A = πr instead of A = πr², gives 4π.
- (c) 2 — First write 1 1/2 as an improper fraction, 3/2. To divide by 3/4, multiply by its reciprocal, 4/3: 3/2 × 4/3 = 12/6 = 2. Dropping the whole number and dividing only the fractional part, 1/2 ÷ 3/4 = 1/2 × 4/3, gives 2/3. Multiplying by 3/4 directly instead of using its reciprocal, 3/2 × 3/4, gives 9/8. Using the reciprocal of the first fraction instead of the second, 2/3 × 3/4, gives 1/2.
- (d) 16π cm² — Method: for a circle, area = π × radius². Working: area = π × 4² = π × 16 = 16π cm². Answer: 16π cm². The student squared the radius but left out the π, which is why 16 cm² is not the exact area. 4π cm² comes from multiplying by the radius once instead of squaring it: π × 4 = 4π. 8π cm² comes from using the circumference formula 2 × π × radius instead of the area formula: 2 × π × 4 = 8π. 64π cm² comes from using the diameter (8 cm) as the radius in the area formula: π × 8² = 64π.
- (a) 1/3 — To subtract these fractions, first write 3/4 with a denominator of 12: 3/4 = 9/12. Then 9/12 − 5/12 = 4/12, which simplifies to 1/3. Subtracting the numerators and the denominators separately, (3 − 5)/(4 − 12), gives −2/−8, which simplifies to 1/4. Changing 3/4 to twelfths by only changing the denominator, without scaling the numerator to match, gives 3/12 − 5/12 = −2/12, which simplifies to −1/6. Adding the fractions instead of subtracting them, 9/12 + 5/12, gives 14/12, which simplifies to 7/6.
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