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AFCAT Numerical Ability · Percentage

Shares of a Whole: Budgets, Savings and Groups

A class, a budget or a salary split into percentage shares: find the rest, take a share of what is left, track savings when income and spending change, and combine groups or overlapping groups.

All Percentage notes

Why this matters

One AFCAT past-year question is here, from 2017: two groups of students with different scholarship rates. Most of the page is EASY, one share and the rest; the marks go on 'of the remaining', which is a share of a share, and on overlapping groups counted twice.

Concept 1 of 6: Parts of a whole, and the rest

If 42% of a school study science and 26% commerce, the remaining 32% study arts. Once one part is known as a number, the whole follows, and so does every other part.

Definition

  • The parts add to 100%. The rest is 100%100\% minus the parts given.
  • A part known as a number fixes the whole: whole =part×100its percentage= \dfrac{\text{part} \times 100}{\text{its percentage}}.
  • 'Passed' and 'failed' are complements: if 30% failed, 70% passed.
  • Answer the part asked, which is often not the part given.

The rest, and the whole

rest=100%−∑parts,whole=100×partp\text{rest} = 100\% - \sum \text{parts}, \qquad \text{whole} = \dfrac{100 \times \text{part}}{p}

Worked example

In a school 42% of students take science and 26% commerce. The other 160 take arts. How many students are there?
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Percentage · Budgets, Savings and Groups

In an exam, 15%15\% students got distinction, 25%25\% got first class, 35%35\% got second class, 20%20\% got third class, and remaining 2020 students failed. Find total students.

Answer the part asked

With 62.5% boys and 30 girls, the class is 80. If the question asks for boys, 80 is the distractor; the answer is 50.
Drill 23 more on parts of a whole, and the rest

Concept 2 of 6: Income, spending and savings

Savings are what is left of an income after spending. If spending takes 78%, savings are 22%, and a savings figure in rupees gives the income the same way as any other part.

Definition

  • Savings = income − expenditure. As percentages, savings % =100−= 100 - spending %.
  • Add up all the spending heads first, then take the rest.
  • Allowances are added on a base: basic pay ₹40,000 with DA 34% and HRA 16% is 40000×1.540000 \times 1.5.
  • Deductions such as tax and PF are taken off the gross: 10% and 6% leave 84%.

Savings from spending

income=savings×100100−spending%\text{income} = \dfrac{\text{savings} \times 100}{100 - \text{spending}\%}

Worked example

A man spends 36% of his income on rent, 24% on food and 18% on travel, and saves the rest, ₹5,500. Find his income.
Practice this conceptself-check · 3 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Percentage · Budgets, Savings and Groups

A person spends 30% of monthly salary on rent, 25% on food, 20% on children's education and 12% on electricity and the balance of ₹1040 on the remaining items. What is the monthly salary of the person?

Divide by the savings share, not the spending

If spending is 78% and savings ₹5,500, the income is 5500÷0.225500 \div 0.22. Dividing by 0.78 treats the savings as the spending.
Drill 19 more on income, spending and savings

Concept 3 of 6: A share of what is left

'25% to his mother and 40% of the rest to his wife' takes the 40% from the 75% left, not from the whole salary. Each step keeps a fraction of what remains, so the fractions multiply.

Definition

  • 'Of the rest' or 'of the remaining' means the next percentage acts on what is left, not on the original.
  • What is finally left is the product of the kept fractions: keep 75% and then 60% of that, so 0.75×0.6=0.450.75 \times 0.6 = 0.45.
  • Divide the amount left by that product to get the original.

Amount left after shares of the rest

left=S(1−a100)(1−b100)\text{left} = S\left(1 - \tfrac{a}{100}\right)\left(1 - \tfrac{b}{100}\right)

Worked example

A man gives 25% of his salary to his mother and 40% of the rest to his wife. He has ₹13,500 left. Find his salary.
Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionEasy

Example 3 · Percentage · Budgets, Savings and Groups

Ranjan spends 20% of his income on house rent and 70% of the remaining on domestic expenditure. If he saves ₹3600 per month, then what is his monthly salary?

'Of the rest' is not of the whole

25% and then 40% of the rest leaves 45%, not 35%. Adding the two percentages treats the 40% as a share of the whole salary.
Drill 3 more on a share of what is left

Concept 4 of 6: When income and spending both change

Savings are a small difference of two big numbers, so they swing much more than either. Take the income as 100, apply each change, and subtract.

Definition

  • Take income 100. Spending s%s\% makes spending ss and savings 100−s100 - s.
  • Apply the income change and the spending change separately, then subtract to get the new savings.
  • Compare the new savings with the old ones for a percentage change, or with the new income for a savings rate.

New savings

new savings=100(1+a100)−s(1+b100)\text{new savings} = 100\left(1 + \tfrac{a}{100}\right) - s\left(1 + \tfrac{b}{100}\right)

Worked example

A man spends 80% of his income. His income rises by 20% and his spending by 10%. By what percent do his savings rise?
Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionModerate

Example 4 · Percentage · Budgets, Savings and Groups

The monthly income and monthly expenditure of Rahul are ₹13500 and ₹9000, respectively. His income and expenditure are increased by 14% and 7%, respectively. Find the percentage increase in his savings.

Savings swing more than income

With income up 20% and spending up 10%, savings rise 60%, not 10% or 20%. Work it out on 100; never guess from the two percentages.
Drill 5 more on when income and spending both change

Concept 5 of 6: Combining groups with different percentages

If boys and girls are in the ratio 3 : 1 and 40% of boys and 60% of girls play a sport, the overall share is not the average of 40 and 60. The bigger group counts more.

Definition

  • Take the groups as their sizes (or ratio parts). Find each group's count with the property, add, and divide by the total.
  • Overall % =n1p1+n2p2n1+n2= \dfrac{n_1 p_1 + n_2 p_2}{n_1 + n_2}: a weighted average.
  • The answer always lies between the two group percentages, nearer the larger group's.
  • Two salaries with different raises work the same way: weight each raise by its salary.

Weighted percentage

overall %=n1p1+n2p2n1+n2\text{overall } \% = \dfrac{n_1 p_1 + n_2 p_2}{n_1 + n_2}

Worked example

Boys and girls in a school are in the ratio 3 : 1. 40% of the boys and 60% of the girls play a sport. What percentage of all students play?
Practice this conceptself-check · 2 quick reps

The same idea in a real exam question:

AFCAT · 2017 · AFCAT (II) 2017 — Numerical Ability · 03-208Easy

Example 5 · Percentage · Budgets, Savings and Groups

The ratio of boys and girls in a school is 3 : 2. 20% of the boys and 25% of the girls have scholarships. What is the total percentage of students who have scholarships?

Do not average the percentages

With boys and girls 3 : 1, 40% and 60% give 45% overall, not 50%. A plain average is right only when the groups are equal.
Drill 6 more on combining groups with different percentages

Concept 6 of 6: Overlapping groups: failed in one, both or neither

Students who failed both subjects are counted in both failure figures. Add the two figures, take the overlap off once, and you have everyone who failed at least one.

Definition

  • At least one =A+B−both= A + B - \text{both}.
  • Neither =100%−= 100\% - at least one. 'Passed in both' is 'failed in neither'.
  • Given 'neither', work backwards: at least one is 100%−100\% - neither, and both =A+B−= A + B - at least one.
  • The same rule works for liking two fruits, reading two papers or owning two things.

Two overlapping groups

n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)

Worked example

In an exam, 40% failed in Hindi, 32% failed in English and 12% failed in both. What percentage passed in both?
Practice this conceptself-check · 2 quick reps

The same idea in a practice question:

Practice questionHard

Example 6 · Percentage · Budgets, Savings and Groups

In an examination, 65%65\% candidates passed in English and 70%70\% passed in Mathematics. If 50%50\% passed in both subjects, what percentage failed in both?

Take the overlap off once

40% and 32% failing with 12% in both is 60% failing at least one. Adding 40 and 32 counts the 12% twice.
Drill 4 more on overlapping groups: failed in one, both or neither

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (6)

  • Parts of a whole, and the rest

    The rest, and the whole

    rest=100%−∑parts,whole=100×partp\text{rest} = 100\% - \sum \text{parts}, \qquad \text{whole} = \dfrac{100 \times \text{part}}{p}
  • Income, spending and savings

    Savings from spending

    income=savings×100100−spending%\text{income} = \dfrac{\text{savings} \times 100}{100 - \text{spending}\%}
  • A share of what is left

    Amount left after shares of the rest

    left=S(1−a100)(1−b100)\text{left} = S\left(1 - \tfrac{a}{100}\right)\left(1 - \tfrac{b}{100}\right)
  • When income and spending both change

    New savings

    new savings=100(1+a100)−s(1+b100)\text{new savings} = 100\left(1 + \tfrac{a}{100}\right) - s\left(1 + \tfrac{b}{100}\right)
  • Combining groups with different percentages

    Weighted percentage

    overall %=n1p1+n2p2n1+n2\text{overall } \% = \dfrac{n_1 p_1 + n_2 p_2}{n_1 + n_2}
  • Overlapping groups: failed in one, both or neither

    Two overlapping groups

    n(A∪B)=n(A)+n(B)−n(A∩B)n(A \cup B) = n(A) + n(B) - n(A \cap B)

Watch out for (6)

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