N Number — printable coverage pack

N1 · 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.

1. Work out $\dfrac{3}{5}+0.35$ .[1 mark]
Answer
- $0.95$
Method: Convert $\dfrac{3}{5}$ to $0.6$ . Then $0.6+0.35=0.95$ .

1. A sum of $\pounds 294$ is divided in the ratio $5:2$ . Work out the larger share.[2 marks]
Answer
- $\pounds 210$
Method: There are $5+2=7$ equal parts. One part is $294\div7=42$ , so the larger share is $5\times42=210$ .

1. 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.[4 marks]
Answer
- $\pounds 160$
Method: The multiplier for a $12\%$ increase is $1.12$ . The original fund was $403.20\div1.12=360$ . The ratio has $5+4=9$ parts, so one part is $360\div9=40$ . The smaller share is $4\times40=160$ .

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.

1. A meal has a choice of $4$ main courses and $3$ desserts. How many different main-course-and-dessert meals are possible?[1 mark]
Answer
- $12$ meals
Method: Each of the $4$ main courses can be paired with any of the $3$ desserts, so the product rule gives $4\times3=12$ .

1. 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.[2 marks]
Answer
- $60$ codes
Method: There are $5$ choices for the first position, $4$ for the second and $3$ for the third. Hence there are $5\times4\times3=60$ codes.

1. 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.[3 marks]
Answer
- $66$ sandwiches
Method: Without the restriction there are $4\times6\times3=72$ sandwiches. With rye bread, the two unavailable fillings would each pair with $3$ sauces, giving $1\times2\times3=6$ forbidden sandwiches. Therefore $72-6=66$ are available.

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.

1. Simplify $\sqrt{72}$ .[2 marks]
Answer
- $6\sqrt{2}$
Method: Use the largest square factor: $72=36\times2$ . Therefore $\sqrt{72}=\sqrt{36}\sqrt{2}=6\sqrt{2}$ .

1. Rationalise the denominator of $\dfrac{5}{\sqrt{7}-2}$ .[3 marks]
Answer
- $\dfrac{5\sqrt{7}+10}{3}$
Method: Multiply by the conjugate: $\dfrac{5}{\sqrt{7}-2}\times\dfrac{\sqrt{7}+2}{\sqrt{7}+2}=\dfrac{5\sqrt{7}+10}{7-4}=\dfrac{5\sqrt{7}+10}{3}$ .

1. Work out the exact value of $\dfrac{3+\sqrt{5}}{\sqrt{5}-1}-\sqrt{5}$ .[4 marks]
Answer
- $2$
Method: Rationalise the fraction using $\sqrt{5}+1$ . Its numerator becomes $(3+\sqrt{5})(\sqrt{5}+1)=8+4\sqrt{5}$ and its denominator becomes $5-1=4$ . The fraction is therefore $2+\sqrt{5}$ , so subtracting $\sqrt{5}$ gives $2$ .