Surds, Bounds & Advanced Number

Simplify √48 to the form a√b, where b has no square factors.

• A16√3
• B2√12
• C4√3✓
• D√48

Answer: 48 splits into the square factor 16 and 3, so √48 = √16 × √3. The square root of 16 is 4, giving 4√3 in simplest form. Leaving the surd as √48 or writing 2√12 has not pulled out the largest square factor, and 16√3 wrongly keeps 16 outside the root instead of square-rooting it.

Simplify √75 − √12 to a single surd term.

• A√2
• B√6
• C3√3✓
• D√3

Answer: √75 simplifies to 5√3 and √12 simplifies to 2√3, since 75 = 25×3 and 12 = 4×3. Subtracting the coefficients gives 5√3 − 2√3 = 3√3. √3 drops the coefficient of 3 entirely after subtracting, while √6 and √2 both come from pairing the original numbers with the wrong square factor.

Simplify √20 × √5.

• A50
• B15
• C10✓
• D25

Answer: Multiplying surds means multiplying the numbers underneath the root, so √20 × √5 = √100, and the square root of 100 is exactly 10, with no surd left over. Fifteen and twenty-five both come from arithmetic slips when combining 20 and 5 under the root, and 50 comes from halving 100 instead of taking its square root.

Simplify (√9)² + (√16)².

• A13
• B19
• C25✓
• D34

Answer: Squaring a square root cancels the root, so (√9)² = 9 and (√16)² = 16. Adding these whole numbers gives 9 + 16 = 25. Thirteen and nineteen both come from square-rooting one term but forgetting to square the other back up, and thirty-four comes from an arithmetic slip when adding.

Write √72 in the form a√b, where a and b are integers and b is as small as possible.

• A√3
• B6√2✓
• C√8
• D√2

Answer: 72 has 36 as its largest square factor, since 36 × 2 = 72, so √72 = √36 × √2 = 6√2. This is fully simplified because 2 has no square factors left. √2 drops the coefficient of 6 entirely, while √3 and √8 both come from pairing 72 with the wrong square factor.

Rationalise the denominator: 8/√2.

• A8√2
• B4√2✓
• C8/√2
• D2√2

Answer: Multiplying top and bottom by √2 gives 8√2 over 2, and dividing by 2 gives 4√2, which has no surd left on the bottom. Writing 8√2 forgets to divide by 2 after multiplying, 2√2 divides the coefficient incorrectly, and 8/√2 is simply the original expression left unrationalised.

Rationalise the denominator: 3/√2.

• A3√2
• B3√2/2✓
• C√2/3
• D3/2

Answer: Multiplying top and bottom by √2 turns the denominator into 2, since √2 × √2 = 2, giving 3√2/2. This has no surd left on the bottom, so it is fully rationalised. Writing 3√2 forgets to divide by 2, 3/2 wrongly drops the surd altogether, and √2/3 mistakenly swaps which number sits on top of the fraction.

Rationalise the denominator: 1/(√7 − 2).

• A(√7 + 2)/3✓
• B√7 + 2
• C(√7 − 2)/3
• D(√7 + 2)/11

Answer: Multiplying top and bottom by the conjugate (√7 + 2) turns the denominator into (√7)² − 2² = 7 − 4 = 3, giving (√7 + 2)/3. Keeping the minus sign from the original denominator is a sign error, forgetting to divide by 3 leaves the numerator alone, and squaring 7 and 2 the wrong way gives an incorrect denominator of 11.

Rationalise the denominator: (√3 + 1)/(√3 − 1).

• A2 + 2√3
• B2 + √3✓
• C4 + 2√3
• D1 + 2√3

Answer: Multiplying top and bottom by the conjugate (√3 + 1) makes the denominator (√3)² − 1² = 2, and the numerator expands to 4 + 2√3, which divides by 2 to give 2 + √3. All three alternatives forget to divide the whole numerator by 2, and the constant term is also miscopied in two of them, leaving 4 + 2√3, 2 + 2√3 or 1 + 2√3 instead of the fully divided answer.

A circle has radius 4 cm. Give its exact area in terms of π.

• A32π cm²
• B8π cm²
• C4π cm²
• D16π cm²✓

Answer: The area of a circle is πr², and squaring the radius gives 4² = 16, so the exact area is 16π cm². Doubling the radius instead of squaring it gives 8π cm², forgetting to square it at all gives 4π cm², and confusing the area formula with π × diameter × radius gives 32π cm².

A cylinder has volume 45π cm³ and cross-sectional area 9π cm². Find its exact height.

• A5π cm
• B5 cm✓
• C9 cm
• D45 cm

Answer: Height equals volume divided by cross-sectional area, and the π terms cancel exactly: 45π ÷ 9π = 5. Leaving a π in the answer forgets that it cancels, using the volume value alone ignores the division, and using the area value alone confuses it with the height.

Simplify exactly: 2√3 × 3√12.

• A36✓
• B18
• C24
• D30

Answer: Multiplying the coefficients gives 2 × 3 = 6, and multiplying inside the roots gives √3 × √12 = √36 = 6, so the product is 6 × 6 = 36. Thirty comes from adding the coefficients instead of multiplying them, 24 comes from a slip in the final multiplication of 6 × 6, and 18 comes from halving the correct answer by mistake.

Without a calculator, between which two consecutive integers does √70 lie?

• A8 and 9✓
• B6 and 7
• C7 and 8
• D9 and 10

Answer: 8² = 64 and 9² = 81, and 70 lies between these two square numbers, so √70 lies between 8 and 9. The pair 7 and 8 uses 7² = 49, which is too small, and 9 and 10 uses 10² = 100, which is too large, so neither bracket contains 70.

Estimate the value of 7.9³ to the nearest 10 by rounding 7.9 to 1 significant figure first.

• A730
• B340
• C510✓
• D640

Answer: Rounding 7.9 to 1 significant figure gives 8, and 8³ = 512, which rounds to 510 to the nearest 10. Rounding down to 7 first and cubing gives 343, rounding to 340, while rounding up to 9 and cubing gives 729, rounding to 730, both of which use the wrong rounded starting value.

A length is measured as 12 cm to the nearest cm. Write down the upper bound of the length.

• A13 cm
• B11.5 cm
• C12.4 cm
• D12.5 cm✓

Answer: A measurement rounded to the nearest whole unit can be up to half a unit higher before it would round up further, so the upper bound of 12 cm is 12.5 cm. 12.4 cm stops just short of the true half-unit boundary, 13 cm rounds a whole unit too far, and 11.5 cm is the lower bound rather than the upper bound.

A mass is recorded as 8.4 kg to 1 decimal place. State the lower bound.

• A8.35 kg✓
• B8.4 kg
• C8.3 kg
• D8.45 kg

Answer: A value rounded to 1 decimal place could have been up to half of 0.1 smaller before rounding, so the lower bound of 8.4 kg is 8.4 − 0.05 = 8.35 kg. 8.3 kg subtracts a whole 0.1 rather than half of it, 8.4 kg repeats the rounded value with no adjustment, and 8.45 kg is the upper bound rather than the lower bound.

Two lengths are measured as 15 cm and 9 cm, each to the nearest cm. Work out the upper bound of their difference.

• A6 cm
• B5 cm
• C24 cm
• D7 cm✓

Answer: The upper bound of a difference uses the larger upper bound minus the smaller lower bound: 15.5 cm − 8.5 cm = 7 cm. 24 cm comes from adding the upper bound of one length to the lower bound of the other instead of subtracting them, and 6 cm and 5 cm both come from pairing the wrong combination of bounds together when subtracting.

A rectangle has sides measured as 6.2 cm and 3.5 cm, each to 1 decimal place. Calculate the upper bound of its area, giving your answer to 3 significant figures.

• A22.0 cm²
• B22.2 cm²✓
• C21.9 cm²
• D21.7 cm²

Answer: The upper bound of the area comes from multiplying the upper bounds of both sides: 6.25 cm × 3.55 cm = 22.1875 cm², which rounds to 22.2 cm² to 3 significant figures. 21.7 cm² simply multiplies the rounded measurements with no bound adjustment at all, 22.0 cm² takes the upper bound of the 3.5 cm side only, and 21.9 cm² takes the upper bound of the 6.2 cm side only.

Speed is calculated as distance ÷ time, where distance = 240 m (to the nearest 10 m) and time = 15 s (to the nearest second). Find the upper bound of the speed, to 1 decimal place.

• A16.3 m/s
• B16.9 m/s✓
• C16.0 m/s
• D15.2 m/s

Answer: The upper bound of a quotient uses the larger upper bound divided by the smaller lower bound: 245 m ÷ 14.5 s = 16.9 m/s to 1 decimal place. 16.0 m/s simply divides the unrounded measurements, 16.3 m/s divides the upper distance by the unrounded time, and 15.2 m/s wrongly divides a lower bound of distance by an upper bound of time.

Convert 0.7̇ (0.777...) to a fraction in its simplest form.

• A7/99
• B7/90
• C7/10
• D7/9✓

Answer: Letting x = 0.7777... and multiplying by 10 gives 10x = 7.7777..., and subtracting x cancels the recurring part to leave 9x = 7, so x = 7/9. 7/10 treats the decimal as if it terminated rather than recurred, while 7/90 and 7/99 both come from using the wrong power of 10 when setting up the subtraction.

Convert 0.4̇5̇ (0.454545...) to a fraction in its simplest form.

• A9/20
• B5/11✓
• C5/9
• D4/9

Answer: Letting x = 0.454545... and multiplying by 100 (since two digits recur) gives 100x = 45.454545..., and subtracting x leaves 99x = 45, so x = 45/99, which simplifies to 5/11. 9/20 treats the decimal as if it terminated at two places, and 4/9 and 5/9 both use a one-digit recurring method on a two-digit repeating block.

Convert 0.1̇2̇ (0.121212...) into a fraction, then add 1/3 to it. Give your answer as a single fraction in simplest form.

• A5/11✓
• B19/33
• C7/22
• D4/11

Answer: 0.1212... converts to 12/99, which simplifies to 4/33, and adding 1/3 (equivalent to 11/33) gives 15/33, which simplifies to 5/11. 4/11 stops after converting the recurring decimal and forgets to add 1/3, while 19/33 and 7/22 both come from adding the two fractions incorrectly before simplifying.