Civil Service Numerical Reasoning Test Percentages, Ratios and Proportional Reasoning Syllabus

In a ratio division, how do you find the total quantity given the value of one part?

Find the value of one share by dividing the known part's value by its number of ratio shares, then multiply by the total number of shares. E.g. if $3$ shares $= \pounds 45$, one share $= \pounds 15$ and a $2:3$ split totals $5 \times 15 = \pounds 75$.

What is the difference between part-to-part and part-to-whole ratios?

A part-to-part ratio compares one part with another (e.g. boys to girls $3:2$). A part-to-whole ratio (often a fraction) compares one part with the total (e.g. boys are $\frac{3}{5}$ of the class).

If the ratio of red to blue is $3:2$, what fraction of the total is red?

$\frac{3}{3+2} = \frac{3}{5}$ of the total is red, and $\frac{2}{5}$ is blue.

How do you convert a part-to-whole fraction into a part-to-part ratio?

If a part is $\frac{a}{b}$ of the whole, the remaining part is $\frac{b-a}{b}$, giving the part-to-part ratio $a:(b-a)$. E.g. $\frac{3}{5}$ gives $3:2$.

How do you compare two ratios to see which is larger?

Write each as a single fraction or convert to the form $1:n$ (or a common second term), then compare. For example, $3:4 = 0.75$ and $5:7 \approx 0.714$, so $3:4$ is the larger ratio.

How do you combine two ratios that share a common term, e.g. $A:B = 2:3$ and $B:C = 4:5$?

Scale so the shared term ($B$) matches: make $B = 12$ in both, giving $A:B = 8:12$ and $B:C = 12:15$. Then $A:B:C = 8:12:15$.

What is the unitary method?

A technique that first finds the value of one unit, then multiplies up to the required number of units. It is the basis of most proportion calculations.

Using the unitary method: if $5$ pens cost \pounds 3.50, what do $8$ pens cost?

One pen costs $\frac{3.50}{5} = \pounds 0.70$; eight pens cost $8 \times 0.70 = \pounds 5.60$.

What characterises direct proportion between two quantities $x$ and $y$?

They increase and decrease in the same ratio, so $y = kx$ for a constant $k$, and $\frac{y}{x}$ is constant. Doubling $x$ doubles $y$.

How do you find the constant of proportionality $k$ in direct proportion?

Use a known pair of values: $k = \frac{y}{x}$. Then $y = kx$ can be used for any other value.

If $y$ is directly proportional to $x$ and $y = 12$ when $x = 4$, find $y$ when $x = 7$.

$k = \frac{12}{4} = 3$, so $y = 3x = 3 \times 7 = 21$.

What characterises inverse proportion between two quantities $x$ and $y$?

As one increases the other decreases so that their product is constant: $xy = k$, i.e. $y = \frac{k}{x}$. Doubling $x$ halves $y$.

If $4$ workers take $6$ days to do a job, how long do $3$ workers take (inverse proportion)?

Total work $= 4 \times 6 = 24$ worker-days (the constant). With $3$ workers, time $= \frac{24}{3} = 8$ days.

How do you decide whether a problem is direct or inverse proportion?

If more of one means more of the other (e.g. more items, more cost) it is direct; if more of one means less of the other (e.g. more workers, less time) it is inverse.

State the formula linking speed, distance and time.

$\text{speed} = \frac{\text{distance}}{\text{time}}$, which rearranges to $\text{distance} = \text{speed} \times \text{time}$ and $\text{time} = \frac{\text{distance}}{\text{speed}}$.

A car travels $150$ km in $2.5$ hours. What is its average speed?

$\text{speed} = \frac{150}{2.5} = 60$ km/h.

How do you convert a time of $2$ hours $30$ minutes into hours for a speed calculation?

Convert minutes to a fraction of an hour: $30$ minutes $= \frac{30}{60} = 0.5$ h, so the time is $2.5$ hours.

How do you convert a speed from km/h to m/s?

Multiply by $\frac{1000}{3600} = \frac{5}{18}$. For example, $36$ km/h $= 36 \times \frac{5}{18} = 10$ m/s.

What is a rate of consumption, and how is it calculated?

The amount of a resource used per unit of time or output, calculated as $\text{rate} = \frac{\text{amount used}}{\text{time or units}}$. E.g. fuel use in litres per hour.

A machine produces $480$ items in $8$ hours. What is its production rate?

$\text{rate} = \frac{480}{8} = 60$ items per hour.

A car uses $45$ litres of fuel to travel $540$ km. What is its fuel consumption in km per litre?

$\frac{540}{45} = 12$ km per litre.

What is a unit (cost) rate, and how do you use it to compare value for money?

It is the cost per single unit, found as $\frac{\text{total cost}}{\text{number of units}}$. The option with the lower cost per unit is better value. E.g. compare price per $100$ g.

Which is better value: $6$ items for \pounds 4.20 or $10$ items for \pounds 6.50?

Unit costs are $\frac{4.20}{6} = \pounds 0.70$ and $\frac{6.50}{10} = \pounds 0.65$. The pack of $10$ is cheaper per item, so it is better value.

How do you calculate the total cost from a unit price and a quantity?

$\text{Total cost} = \text{unit price} \times \text{quantity}$. E.g. $\pounds 2.30$ per kg for $4$ kg gives $\pounds 9.20$.

What is an exchange rate?

The price of one currency expressed in terms of another, e.g. \pounds 1 = \$1.25, telling you how many units of the second currency one unit of the first will buy.

How do you convert from your home currency to a foreign currency given a rate?

Multiply by the exchange rate. If \pounds 1 = \$1.25, then \pounds 80 = $80 \times 1.25 = \$100$.

How do you convert from a foreign currency back to your home currency?

Divide by the exchange rate. If \pounds 1 = \$1.25, then \$150 = $\frac{150}{1.25} = \pounds 120$.

Given \pounds 1 = \euro 1.15, convert \euro 92 into pounds.

Divide by the rate: $\frac{92}{1.15} = \pounds 80$.

How do you find a cross-rate, e.g. dollars to euros, given \pounds 1 = \$1.25 and \pounds 1 = \euro 1.15?

Divide the two rates: $\$1 = \frac{1.15}{1.25} = \euro 0.92$ (euros per dollar), or equivalently $\euro 1 = \frac{1.25}{1.15} \approx \$1.087$.

How do you apply two successive percentage changes, such as a $10\%$ rise followed by a $20\%$ fall?

Multiply the multipliers together: $1.10 \times 0.80 = 0.88$, an overall $12\%$ decrease. Successive changes do not simply add.