OxfordAQA International GCSE Mathematics: Rounding, estimation, bounds and fraction-decimal equivalence (9260) – Revision Notes

These revision notes condense the OxfordAQA International GCSE Mathematics (9260) specification, Version 5.1 (for exams May/June 2018 onwards), statements N11 (section 3.1.1) and N12 to N14 (section 3.1.2). All content can be assessed on the Extension Tier question papers. Items marked “Extension only” are in the Extension column of the specification only. For full explanations, read the study guide.

Links: course hub, printable checklist, practice questions and the free diagnostics.

Key definitions

Term	Meaning
Decimal place	A digit after the decimal point, counted from the point
Significant figure	A digit, counting from the first one that is not zero
Error interval	The range of true values that round to a given value
Lower bound	The bottom of the error interval (included)
Upper bound	The top of the error interval (not reached, but used in calculations)
Terminating decimal	A decimal that ends
Recurring decimal	A decimal with a repeating digit or block (Extension only)
Operator	A fraction, decimal or percentage that multiplies a quantity

Rounding (N11)

Method in steps:

1. Mark the last digit you are keeping.
2. Look at the digit just after it. 5 or more: round up. 4 or less: leave it.
3. Replace dropped digits before the decimal point with zeros so the size is right.
4. Keep any trailing zeros that show the accuracy, such as 4.30 to 2 d.p.

Reminder: 0.003951 to 2 s.f. is 0.0040, because the first two significant figures are 3 and 9, and the next digit is 5, so 3.9 rounds up to 4.0. The trailing zero is needed.

Limits of accuracy (N11)

Error interval = value ± half of the rounding unit. Use ≤ at the bottom and < at the top.

Statement	Rounding unit	Error interval
530, nearest 10	10	525 ≤ x < 535
0.8, 1 s.f.	0.1	0.75 ≤ x < 0.85
2.30 kg, 3 s.f.	0.01	2.295 ≤ m < 2.305
45 000, 2 s.f.	1 000	44 500 ≤ n < 45 500

Check your last answer with a test value: does 2.305 round to 2.31? Yes, so it must be excluded. Also check the lower end: 2.295 rounds to 2.30, so it is included.

To read an error interval the other way, ask whether a given number would round to the stated value. For 530 to the nearest 10, a true value of 534.9 is possible and a true value of 535 is not.

Estimation (N11)

Method in steps:

1. Round each number to one significant figure, or to a value that makes the arithmetic easy.
2. Show the rounded numbers.
3. Do the calculation with them.
4. If asked, say whether the result is an over-estimate or an under-estimate.

Direction rule: rounding a numerator up raises the answer, but rounding a denominator up lowers it. Mixed directions mean you cannot tell without checking.

Reminder: 9.7 × 0.52 ÷ 4.9 becomes 10 × 0.5 ÷ 5 = 1. The exact value is 1.03 to 3 s.f.

Bounds (N11, Extension only)

Calculation	Upper bound	Lower bound
a + b	upper a + upper b	lower a + lower b
a − b	upper a − lower b	lower a − upper b
a × b	upper a × upper b	lower a × lower b
a ÷ b	upper a ÷ lower b	lower a ÷ upper b

Method in steps:

1. Write the error interval for each rounded value.
2. Decide which end of each interval pushes the answer the way you want.
3. Calculate with unrounded bounds.
4. To state an answer to a justified accuracy, round both bounds together and keep the greatest accuracy at which they still agree.

Worked reminder. p = 150 (nearest 10) and q = 0.6 (1 d.p.). Find the bounds of p ÷ q.

145 ≤ p < 155        0.55 ≤ q < 0.65
upper = 155 ÷ 0.55 = 281.8...
lower = 145 ÷ 0.65 = 223.1...

To 1 s.f., the bounds are 300 and 200. They disagree, so p ÷ q cannot be given even to 1 s.f. from this data.

Equivalent fractions and conversions (N12)

From	To	Method
Fraction	Decimal	numerator ÷ denominator
Decimal	Fraction	digits over 10, 100, 1 000, then simplify
Decimal	Percentage	× 100
Percentage	Decimal	÷ 100
Percentage	Fraction	over 100, then simplify

Common set to remember:

Fraction	Decimal	Percentage
1/8	0.125	12.5%
3/8	0.375	37.5%
5/8	0.625	62.5%
1/20	0.05	5%
3/40	0.075	7.5%

Reminder: 5/16 = 0.3125 = 31.25%, and 0.65 = 65/100 = 13/20.

A fraction in lowest terms terminates if its denominator has only 2 and 5 as prime factors. Otherwise it recurs.

Recurring decimals (N12, Extension only)

Dot notation: put a dot over each end of the repeating block. 0.1̇2̇ = 0.121212… and 0.16̇ = 0.16666…

Decimal to fraction, in steps:

1. Let x = the decimal.
2. Multiply by 10, 100 or 1 000 so the repeating part is the same after the point in two lines.
3. Subtract the lines to remove the repeat.
4. Solve for x and simplify.

x = 0.121212...
100x = 12.121212...
99x = 12
x = 12/99 = 4/33

Fraction to decimal: divide, and stop when a remainder repeats. For example, 1/6 = 0.16̇.

The number of 9s in the denominator matches the length of the repeating block: one repeating digit gives 9, two give 99, and so on.

Operators (N13)

Operator	How to apply it
a/b of Q	Q ÷ b × a
p% of Q	Q × p/100
0.35 of Q	Q × 0.35
Operator above 1	The result is larger than Q

Reminder: 3/7 of 91 = 91 ÷ 7 × 3 = 39. Chained operators are applied from the right: 1/2 of 3/4 of 80 is 1/2 of 60 = 30.

One quantity as a fraction or percentage of another (N14)

Method in steps:

1. Put both quantities in the same unit.
2. Write the first over the second (the one after “of”).
3. Simplify the fraction, or multiply by 100 for a percentage.
4. Results greater than 1, or greater than 100%, are acceptable.

Reminder: 18 minutes of 1 hour = 18/60 = 3/10 = 30%. For 85 cm compared with 2.5 m, change 2.5 m to 250 cm first: 85/250 = 17/50 = 34%. 420 as a percentage of 300 = 420 ÷ 300 × 100 = 140%.

Must-know distinctions

• Decimal places count from the point. Significant figures begin at the first digit that is not zero.
• Lower bound is included in an error interval. Upper bound is not included, though it is used in calculations.
• “A as a percentage of B” has B on the bottom. Reversing the roles gives a different answer.
• For a − b the upper bound uses the lower bound of b.
• Terminating and recurring decimals are both rational, so both can be written as fractions.

Quick self-test

1. Round 0.003951 to 2 s.f.
2. Round 46.0953 to 2 d.p.
3. Give the error interval for 530 measured to the nearest 10.
4. Give the error interval for 0.8 given to 1 s.f.
5. Estimate 9.7 × 0.52 ÷ 4.9.
6. (Extension only) m = 18 correct to the nearest whole number and n = 4.5 correct to 1 d.p. Find the upper bound of m ÷ n to 3 s.f.
7. (Extension only) With the same values, find the lower bound of m + n.
8. Write 0.65 as a fraction in lowest terms.
9. (Extension only) Write 0.4̇5̇ as a fraction in lowest terms.
10. Write 5/16 as a decimal and as a percentage.
11. Find 3/7 of 91.
12. Write 420 as a percentage of 300.

Answers

1. 0.0040
2. 46.10
3. 525 ≤ x < 535
4. 0.75 ≤ x < 0.85
5. 10 × 0.5 ÷ 5 = 1
6. 18.5 ÷ 4.45 = 4.16 to 3 s.f.
7. 17.5 + 4.45 = 21.95
8. 13/20
9. x = 0.454545…, 99x = 45, x = 45/99 = 5/11
10. 0.3125 and 31.25%
11. 39
12. 140%

Where marks are usually lost

• Writing 46.1 when 2 d.p. is asked for, which loses the final zero.
• Writing the lower end of an error interval with < instead of ≤.
• Using a 3 s.f. interval (14 950 to 15 050) for 15 000 given to 2 s.f., which should be 14 500 to 15 500.
• Rounding intermediate values in a bounds calculation before the final step.
• Using upper bounds for both values in a subtraction or division.
• Giving a final answer to more accuracy than the bounds support.
• Not showing the rounded numbers in an estimation.
• Subtracting the two sides of a recurring decimal incorrectly, so that a repeat is left in.
• Reversing the roles in a “fraction of another” question.
• Forgetting to change hours to minutes (or metres to centimetres) first.

Official syllabus

OxfordAQA International GCSE Mathematics (9260), specification Version 5.1, for exams May/June 2018 onwards, published by OxfordAQA. Sections 3.1.1 (N11) and 3.1.2 (N12, N13, N14).