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

Percentages, Ratios and Proportional Reasoning syllabus — full chapter and topic list

Expand any chapter to see its topics and sub-topics. This is the whole examinable outline for Percentages, Ratios and Proportional Reasoning in Civil Service Numerical Reasoning Test, not a summary of it.

1. 1

   Percentage Calculations

   4 topics

   • Finding a percentage of a quantity
     • Using 10% and 1% building blocks
     • Non-round percentages (e.g. 17.5%, 23%)
   • Percentage increase and decrease
     • Multiplier method (×1.15, ×0.85)
     • Successive percentage changes
   • Reverse percentages
     • Finding the original value before a change
     • Removing VAT or a discount
   • Expressing one figure as a percentage of another
     • Share of a total
     • Percentage of target achieved
2. 2

   Ratio and Apportionment

   4 topics

   • Simplifying and scaling ratios
     • Equivalent ratios and unit form
   • Dividing a quantity in a given ratio
     • Sharing budgets or staff across teams
     • Finding one part given another
   • Combining and comparing ratios
     • Ratios across multiple categories
   • Part-to-part versus part-to-whole
3. 3

   Direct and Inverse Proportion

   3 topics

   • Unitary method
     • Scaling recipes, costs and quantities
   • Direct proportion problems
     • Cost per unit and bulk pricing
   • Inverse proportion problems
     • Workers and time to complete a task
     • Speed and journey time
4. 4

   Rates and Per-Unit Measures

   4 topics

   • Speed, distance and time
     • Average speed across stages
     • Unit and time conversions in rate problems
   • Rates of consumption and production
     • Output per hour or per worker
   • Cost rates and pricing
     • Price per kilogram, per unit, per head
     • Cost per capita comparisons
   • Exchange rates and currency conversion
     • Converting between two currencies
     • Effect of changing rates

Percentages, Ratios and Proportional Reasoning flashcards for Civil Service Numerical Reasoning Test

24 of 50 cards from the Percentages, Ratios and Proportional Reasoning deck — real questions with worked answers.

1. How do you find a percentage of a quantity? Give the general formula.

   Multiply the quantity by the percentage written as a fraction or decimal: $\text{percentage of quantity} = \frac{p}{100} \times Q$. For example, $15\%$ of $240$ is $\frac{15}{100} \times 240 = 36$.

2. What is a quick mental method to find $10\%$, $5\%$ and $1\%$ of a quantity?

   $10\%$ is the quantity divided by $10$; $5\%$ is half of $10\%$; $1\%$ is the quantity divided by $100$. Build other percentages by adding these, e.g. $16\% = 10\% + 5\% + 1\%$.

3. How do you convert a percentage to a decimal multiplier?

   Divide the percentage by $100$. For example, $37\% \to 0.37$ and $4\% \to 0.04$. This decimal is then multiplied by the quantity.

4. What is the formula for a percentage increase of a quantity?

   $\text{New value} = \text{Original} \times \left(1 + \frac{p}{100}\right)$. For a $20\%$ increase, multiply by $1.20$.

5. What is the formula for a percentage decrease of a quantity?

   $\text{New value} = \text{Original} \times \left(1 - \frac{p}{100}\right)$. For a $15\%$ decrease, multiply by $0.85$.

6. What single multiplier represents an increase of $35\%$?

   $1.35$, since $1 + \frac{35}{100} = 1.35$. Multiplying the original value by $1.35$ gives the increased value.

7. What single multiplier represents a decrease of $8\%$?

   $0.92$, since $1 - \frac{8}{100} = 0.92$.

8. How do you calculate the percentage change between an old and a new value?

   $\text{Percentage change} = \frac{\text{New} - \text{Old}}{\text{Old}} \times 100\%$. A positive result is an increase, a negative result is a decrease.

9. What is a reverse percentage problem, and what is the key idea for solving it?

   It is finding the original value when you know the value after a percentage change. The key idea is to divide by the multiplier rather than multiply: $\text{Original} = \frac{\text{Final}}{\text{multiplier}}$.

10. A price is \pounds 60 after a $20\%$ increase. What was the original price?

    Divide by the multiplier $1.20$: $\frac{60}{1.20} = \pounds 50$. (A common error is taking $20\%$ off \pounds 60.)

11. An item costs \pounds 84 after a $30\%$ discount. What was the original price?

    The sale price is $70\%$ of the original, so $\text{Original} = \frac{84}{0.70} = \pounds 120$.

12. How do you express one figure as a percentage of another?

    $\frac{\text{part}}{\text{whole}} \times 100\%$. For example, $18$ as a percentage of $40$ is $\frac{18}{40} \times 100\% = 45\%$.

13. $32$ out of $50$ marks: what percentage is this?

    $\frac{32}{50} \times 100\% = 64\%$.

14. What does it mean to simplify a ratio, and how is it done?

    Writing it in its smallest whole-number terms by dividing every part by their highest common factor. For example, $12:18$ divided by $6$ gives $2:3$.

15. How do you simplify a ratio that contains fractions or decimals?

    Multiply every part by the same number to clear fractions or decimals, then divide by the highest common factor. For example, $0.5:1.5 \to 1:3$ and $\frac{1}{2}:\frac{1}{3} \to 3:2$.

16. How do you write the ratio $a:b$ in the form $1:n$?

    Divide both parts by $a$: $1:\frac{b}{a}$. For example, $4:10 \to 1:2.5$.

17. What does scaling a ratio mean?

    Multiplying (or dividing) every part of the ratio by the same number to keep the proportion the same. For example, scaling $2:3$ by $5$ gives $10:15$.

18. Describe the process for dividing a quantity in a given ratio.

    Add the ratio parts to get the total number of shares, divide the quantity by that total to find the value of one share, then multiply each ratio part by the value of one share.

19. Divide \pounds 120 in the ratio $2:3$. What does each part receive?

    Total shares $= 2+3 = 5$; one share $= \frac{120}{5} = 24$. Parts are $2 \times 24 = \pounds 48$ and $3 \times 24 = \pounds 72$.

20. Share $450$ in the ratio $4:5:6$. What is each amount?

    Total shares $= 15$; one share $= \frac{450}{15} = 30$. Amounts are $120$, $150$ and $180$.

21. 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 $5 \times 15 = \pounds 75$.

22. 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).

23. 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.

24. 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 $3:2$.

See more Percentages, Ratios and Proportional Reasoning flashcards →

Planning Percentages, Ratios and Proportional Reasoning for Civil Service Numerical Reasoning Test

Percentages, Ratios and Proportional Reasoning (Civil Service Numerical Reasoning Test) FAQ