Number

Section N

3 spec leafs

Notes and three levels of exam-style practice for each registered specification leaf in this section.

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Knowledge and use of numbers and the number system including fractions, decimals, percentages, ratio, proportion and order of operations

Fractions, decimals and percentages are different forms of the same value; convert to the form that makes the calculation simplest.
For a ratio $a:b$ , there are $a+b$ equal parts. Find one part before scaling to either share.
Use brackets, then powers and roots, then multiplication and division, then addition and subtraction; operations at the same level are completed from left to right.
A percentage multiplier above $1$ represents an increase and one below $1$ a decrease. A common error is to use the final amount as the base in a reverse-percentage problem.

Tier 1 · Easy

1 mark

ORIGINAL

Work out $\dfrac{3}{5}+0.35$ .

Tier 2 · Standard

2 marks

ORIGINAL

A sum of $\pounds 294$ is divided in the ratio $5:2$ . Work out the larger share.

Tier 3 · Hard

4 marks

ORIGINAL

After an increase of $12\%$ , a fund contains $\pounds 403.20$ . Its original value is divided in the ratio $5:4$ . Work out the smaller share.

The product rule for counting

When a process has successive independent choices, multiply the number of options at each stage.
A counting diagram or systematic list can identify the stages before the product rule is used.
If a restriction removes outcomes, count all unrestricted outcomes first and then subtract the forbidden cases when that is simpler.
Do not add the numbers of choices for successive stages; addition is used for mutually exclusive alternatives, not for choices made together.

Tier 1 · Easy

1 mark

ORIGINAL

A meal has a choice of $4$ main courses and $3$ desserts. How many different main-course-and-dessert meals are possible?

Tier 2 · Standard

2 marks

ORIGINAL

A code consists of one of $5$ letters, followed by one of $4$ digits, followed by one of $3$ symbols. Repetition is allowed. Work out the number of possible codes.

Tier 3 · Hard

3 marks

ORIGINAL

A sandwich uses one of $4$ breads, one of $6$ fillings and one of $3$ sauces. For rye bread, two of the fillings are unavailable. Work out the number of available sandwiches.

Manipulation of surds, including rationalising the denominator; the use of surds in exact calculations

A surd is an irrational root left in exact form, such as $\sqrt{3}$ ; do not replace it with a rounded decimal unless asked.
Simplify $\sqrt{ab}$ by extracting square factors: $\sqrt{ab}=\sqrt{a}\sqrt{b}$ for non-negative $a$ and $b$ .
Collect only like surds, in the same way that like algebraic terms are collected.
To rationalise a two-term denominator, multiply numerator and denominator by its conjugate; the denominator then uses the difference of two squares.

Tier 1 · Easy

2 marks

ORIGINAL

Simplify $\sqrt{72}$ .

Tier 2 · Standard

3 marks

ORIGINAL

Rationalise the denominator of $\dfrac{5}{\sqrt{7}-2}$ .

Tier 3 · Hard

4 marks

ORIGINAL

Work out the exact value of $\dfrac{3+\sqrt{5}}{\sqrt{5}-1}-\sqrt{5}$ .