Applications: Elections, Exams, Spending and Mixtures

Elections: From the Voter List to the Margin

Election questions look long, but the votes always flow down the same ladder. Each percentage in the question is taken on one of these levels, and most mistakes come from using the wrong one.

A candidate's share is a share of the valid votes unless the question says otherwise.

Two candidates. If the winner gets w% of the valid votes, the loser gets (100 − w)%, and the margin is

$$w - (100 - w) = (2w - 100)\% \text{ of the valid votes}$$

"The winner gets 57% of the valid votes and wins by 4,200 votes." The margin is 57 − 43 = 14% of the valid votes, so the valid votes are 4200 ÷ 0.14 = 30,000.

Walking down the ladder. "A constituency has 1,20,000 voters on its list. 25% of them did not vote, and 4% of the votes cast were invalid. Of the two candidates, the winner got 55% of the valid votes. How many votes did the loser get?"

• Votes polled = 75% of 1,20,000 = 90,000.
• Valid votes = 96% of 90,000 = 86,400.
• Loser's votes = 45% of 86,400 = 38,880.

Walking back up. Turn every step into a multiplier and divide once. If the question had given the loser's 38,880 votes and asked for the list, the answer is 38,880 ÷ (0.75 × 0.96 × 0.45) = 38,880 ÷ 0.324 = 1,20,000.

Shares of the votes polled. Some questions give each candidate's share of all the votes polled, with the invalid votes making up the rest. "A got 45% of the votes polled and B got 40%; the rest were invalid." Then 15% were invalid, and A's margin is 5% of the votes polled. Read carefully which base each figure is on.

"Majority" means margin. "Elected by a majority of 1,200 votes" means the winner got 1,200 more votes than the runner-up. It does not mean more than half of the votes.

Elections: Three Candidates, Minimum Shares and Switched Votes

Three candidates. The three shares add up to 100% of the valid votes. "A gets 42% and B gets 33% of the valid votes, and C gets the rest. A beats C by 6,800 votes. Find the valid votes." C has 25%, so A − C = 17% of the valid votes = 6,800, and the valid votes are 40,000. Check carefully which two candidates the margin is between; using A − B here would give a wrong answer.

"Got x% and lost by k." With two candidates, x% for the loser means (100 − x)% for the winner, so the margin is (100 − 2x)%. A candidate who got 35% of the votes and lost by 9,000 votes was behind by 30%, so there were 30,000 votes.

A minimum share to win. "To win, a candidate needs at least 40% of the votes polled. A candidate gets 31% and falls 2,700 votes short." The shortfall is 40 − 31 = 9% of the votes polled, so the votes polled are 2700 ÷ 0.09 = 30,000.

When a share is of the whole list. The longest Tier 1 version mixes two levels. "5% of the voters on the list did not vote and 200 votes were invalid. Of the two candidates, the winner got the votes of 50% of all the voters on the list and won by 1,200 votes. How many voters were on the list?"

Say N voters are on the list.

• Votes polled = 0.95N, valid votes = 0.95N − 200.
• Winner = 0.5N, so loser = (0.95N − 200) − 0.5N = 0.45N − 200.
• Margin = 0.5N − (0.45N − 200) = 0.05N + 200 = 1,200.

So 0.05N = 1,000 and N = 20,000. Check: winner 10,000 and loser 8,800.

[Tier 2] Switched votes. Each voter who switches takes one vote away from A and gives one to B, so the gap between them changes by 2 for every switched voter.

"A beats B by 8% of the votes polled. If 900 voters had voted for B instead of A, B would have won by 4% of the votes polled. Find the votes polled." The gap swings from A ahead by 8% to B ahead by 4%, a swing of 12% of the votes polled. The switch moves the gap by 2 × 900 = 1,800 votes. So 12% = 1,800 and the votes polled are 15,000. Check: A 8,100 and B 6,900 at first; after the switch A 7,200 and B 7,800.

Votes needed to change the result. In the same election, how many voters would have had to switch just to make it a tie? The gap is 1,200 votes, and each switch closes it by 2, so 600 voters.

[SSC Tip] In switch questions, add the two margins when the winner changes (8% + 4%) and subtract them when the same candidate still wins. Then remember the factor 2.

Examinations: Pass Marks, Failed By and Maximum Marks

Pass marks = pass percentage × maximum marks. Two phrases do all the work:

• "got m marks and failed by f": pass marks = m + f
• "got m marks, k more than the pass marks": pass marks = m − k

"To pass, a student needs 33% of the maximum marks. She gets 185 marks and fails by 46." Pass marks = 185 + 46 = 231, which is 33% of the maximum, so the maximum is 231 ÷ 0.33 = 700.

Two students, one pass mark. "A gets 32% and fails by 24 marks. B gets 41% and gets 30 marks more than the pass mark." Picture a number line: A is 24 below the pass mark and B is 30 above it. So the 9% gap between them is 24 + 30 = 54 marks, and the maximum is 54 ÷ 0.09 = 600. Pass mark = 32% of 600 + 24 = 192 + 24 = 216, which is 36%.

If both students are on the same side, subtract instead of adding. "A gets 30% and fails by 30; B gets 40% and fails by 10." The 10% gap is 30 − 10 = 20 marks, so the maximum is 200 and the pass mark is 60 + 30 = 90.

Grace marks. Grace marks are simply added to the score. "The pass mark is 40%. A student gets 170 and fails by 30. Every student is given 5% of the maximum as grace marks. Does he pass now?" The maximum is 200 ÷ 0.4 = 500, so the grace is 25 marks. His new score is 195, still 5 short of 200.

Marks across papers. The overall percentage is total marks ÷ total maximum, not the average of the percentages (unless every paper has the same maximum). "Papers of 50, 100 and 150 marks; she scores 80%, 70% and 60%." Her marks are 40, 70 and 90, so 200 out of 300, which is $66\frac{2}{3}\%$. The plain average of 80, 70 and 60 is 70%, which is wrong here.

Marks needed in the last paper. "There are five papers of 100 marks each. To get 75% overall, how much does he need in the fifth if he has 70, 68, 82 and 79 in the first four?" He needs 375 in all and has 299, so he needs 76.

Failing in One or Both Subjects

Take the whole class as 100% and use inclusion-exclusion:

$$\text{failed in at least one} = \text{failed in A} + \text{failed in B} - \text{failed in both}$$

and passed in both = 100% − failed in at least one.

"In an exam, 35% failed in Maths, 25% failed in English and 12% failed in both." Failed in at least one = 35 + 25 − 12 = 48%, so 52% passed in both. If 1,300 students passed in both, 52% is 1,300 and the class has 2,500.

It helps to split everyone into four groups:

So "failed in exactly one subject" is 23 + 13 = 36%, and "passed in Maths" is 100 − 35 = 65%.

When pass figures are given. Work with "passed" in the same way. "80% passed in Maths, 75% passed in Science and 65% passed in both." Passed in at least one = 80 + 75 − 65 = 90%, so 10% failed in both.

Finding "both" from the rest. "75% passed in English, 60% passed in Maths and 15% failed in both. If 1,000 students passed in both, how many took the exam?" Passed in at least one = 100 − 15 = 85%. Passed in both = 75 + 60 − 85 = 50%. So 50% is 1,000 and 2,000 took the exam.

"15% failed in Science, 20% failed in Maths and 70% passed in both." Failed in at least one = 30%, so failed in both = 15 + 20 − 30 = 5%.

The same idea in other clothes. Readers of two newspapers, people who like tea and coffee, people who speak Hindi and English: "neither" plays the part of "passed in both" (or "failed in both"), and the formula is unchanged.

[Tier 2] Three subjects. At least one = A + B + C − (each pair) + (all three). "20% failed in A, 15% in B and 10% in C; 5% failed in A and B, 4% in B and C, 3% in C and A, and 2% in all three." At least one = 45 − 12 + 2 = 35%, so 65% passed in all three.

[SSC Tip] Never multiply the two pass percentages to get "passed in both". The "both" figure is what links the two subjects, so always use it or find it.

Salary Spending Chains

The whole topic rests on one rule. "x% of his salary" is taken on the salary. "x% of the remaining" is taken on whatever is left at that point, so each such step multiplies what is left by $\left(1 - \frac{x}{100}\right)$.

"A man spends 20% of his salary on rent, 25% of the remaining on food and 10% of what is left after that on travel."

So he saves 54% of his salary, which is just 0.8 × 0.75 × 0.9 = 0.54. If he saves ₹21,600, the salary is 21,600 ÷ 0.54 = ₹40,000.

Finding a middle amount. Find the salary first, then the item. In the chain above, food is 20% of ₹40,000 = ₹8,000.

Mixed chains. When some items are a % of the salary and others a % of the rest, add up the "of salary" ones first. "She spends 25% of her salary on rent and 15% of her salary on food. Of the rest she spends $\frac{1}{3}$ on loan payments and saves ₹12,000." Rent and food take 40%, leaving 60%. She saves $\frac{2}{3}$ of that, which is 40% of the salary. So the salary is 12,000 ÷ 0.4 = ₹30,000.

Students often mix up the two kinds. If the food line said "15% of the remaining", the saving would be 0.75 × 0.85 × $\frac{2}{3}$ = 42.5% instead, and the answer would change.

Gifts and shares. Money given away works the same way. "A man gives $\frac{1}{4}$ of his money to his son, $\frac{1}{3}$ of the rest to his daughter and 10% of what remains to a temple. He is left with ₹36,000." What is left = $\frac{3}{4} \times \frac{2}{3} \times 0.9$ = 0.45 of his money, so he had 36,000 ÷ 0.45 = ₹80,000.

When the saving comes first. "He saves 10% of his salary and spends 30% of the rest on rent" means the rest is 90% of the salary, and rent is 30% of 90% = 27% of the salary. The order only changes what "the rest" means.

Budgets, Pie Charts and Simple Tables

A budget split. A family's spending and savings together make 100% of its income. Any item in rupees = its % × income, and two items differ by (difference of their %) × income.

If the monthly income is ₹50,000, education costs 15% of it, ₹7,500, and food costs (30 − 12)% = 18% of it, ₹9,000, more than the savings. If instead only one rupee figure is given, use it to find the income: rent of ₹9,600 means the income is 9,600 ÷ 0.2 = ₹48,000.

Pie charts. A full circle is 360° for 100%, so 1% = 3.6° and 1° = $\frac{5}{18}$%. Education's slice above is 15 × 3.6 = 54°. A slice of 72° stands for 72 ÷ 3.6 = 20%.

The overall percentage from a table. "Centre A had 600 candidates and 70% passed. Centre B had 400 and 85% passed." The overall pass percentage is total passed ÷ total appeared.

The answer is 76%, not the plain average of 77.5%. The bigger centre pulls the overall figure towards its own percentage.

Change from one year to the next. A school's pass count goes from 320 to 368. The change is 48 on 320, which is 15%. Data interpretation questions use exactly these steps on bigger tables: read two numbers, take the difference, divide by the old one.

What % of the total. "Of 1,250 students in a school, 450 are in the senior wing." That is 450 ÷ 1,250 × 100 = 36%. Converting the answer to degrees for a pie chart is one more multiplication: 36 × 3.6 = 129.6°.

When the options for an overall percentage are close, check that your answer lies between the group percentages and nearer the bigger group. If it doesn't, something has gone wrong.

Population Groups: Gender, Literacy and Other Splits

Questions on a town's men and women, a class's boys and girls or a company's staff all use the same steps: split the total into groups, take each group's percentage, then add.

"A town has 50,000 people and 60% of them are men. 70% of the men and 55% of the women are literate. Find the literacy rate of the town."

• Men = 30,000, of whom 70% = 21,000 are literate.
• Women = 20,000, of whom 55% = 11,000 are literate.
• Literate = 32,000 out of 50,000, which is 64%.

In one line, the overall rate is a weighted average: 0.6 × 70 + 0.4 × 55 = 42 + 22 = 64%.

Working backwards with alligation. If you know the overall rate and the two group rates, you can find the split. Overall 64%, men 70%, women 55%: men : women = (64 − 55) : (70 − 64) = 9 : 6 = 3 : 2, so 60% are men.

The other part of a group. Illiterate women = 45% of 20,000 = 9,000. Literate men are 21,000 ÷ 32,000 of all literate people, which is 65.625%.

Groups that grow at different rates. "A town has 8,000 people. The number of men goes up 6% and the number of women goes up 10%, and the population becomes 8,600. How many men were there?" If everyone had gone up 6%, the rise would be 480. The actual rise is 600, so the extra 120 comes from the women's extra 4%. Women = 120 ÷ 0.04 = 3,000, and men = 5,000. The equation 0.06m + 0.1(8000 − m) = 600 gives the same.

Scholarships, jobs and votes by group. "40% of the students in a college are girls. 30% of the girls and 20% of the boys get scholarships. What % of the students get none?" Getting one: 0.4 × 30 + 0.6 × 20 = 12 + 12 = 24%. So 76% get none.

Counts from a percentage difference. "In a class, boys are 60% and there are 12 more boys than girls." The 20% gap is 12 students, so the class has 60 students: 36 boys and 24 girls.

Don't just average the two group percentages: (70 + 55) ÷ 2 = 62.5% is wrong for the town above. Weight each one by the size of its group.

Mixtures and Alloys by Percentage

Concentration = part ÷ whole × 100. The key to every question here is to find the part that does not change.

Adding water (or any one part): the other part stays fixed. "40 litres of milk and water has 10% water. How much water must be added to make the water 20%?" Milk = 36 L, and it does not change. In the new mixture milk is 80%, so the new total is 36 ÷ 0.8 = 45 L. Water to add = 5 L.

Evaporation: the dissolved part stays. "50 kg of salt water has 6% salt. How much water must evaporate to make it 10% salt?" Salt = 3 kg. At 10% that is the salt in 30 kg, so 20 kg of water must evaporate.

Fresh and dry fruit: the solid part stays. "Fresh fruit has 75% water and dried fruit has 10% water. How much dried fruit comes from 18 kg of fresh fruit?" Solid = 25% of 18 = 4.5 kg. In the dried fruit this is 90% of the weight, so the dried fruit weighs 4.5 ÷ 0.9 = 5 kg. Answers like 13.5 kg (taking the water percentages straight off) are a sign that the solid part was not held fixed.

Adding a pure metal to an alloy. "A 30 kg alloy has 20% zinc. How much copper must be added so that zinc is 15%?" Zinc = 6 kg and stays. At 15% that means a total of 40 kg, so 10 kg of copper is added.

Mixing two mixtures. Add the parts and add the wholes. "20 L of 30% acid is mixed with 30 L of 10% acid." Acid = 6 + 3 = 9 L in 50 L, which is 18%. This is a weighted average, so alligation gives the mixing ratio for a target: to get 15% from 30% and 10%, mix them in the ratio (15 − 10) : (30 − 15) = 5 : 15 = 1 : 3.

The bucket question. "An empty bucket weighs 20% of the bucket full of water. Some water is poured out, and the bucket with the water left weighs 60% of the full weight. What % of the water was poured out?" Take the full weight as 100: bucket 20, water 80. Now the total is 60, so the water is 40. Half the water, 50%, was poured out.

Taking out part of a mixture and topping it up with water several times is a different method, covered in Mixture and Alligation.

Commission, Tax, GST and Other Everyday Percentages

Commission. A salesman's commission is a percentage of his sales. An agent on 6% who sells goods worth ₹1,25,000 earns ₹7,500.

With slabs, each rate applies only to its own slice of the sales. "A salesman gets 5% commission on the first ₹10,000 of sales and 4% on sales above that. He keeps his commission and sends ₹47,900 to the company. Find his total sales." Say the sales are S. Commission = 500 + 0.04(S − 10,000) = 0.04S + 100. He sends S − 0.04S − 100 = 0.96S − 100 = 47,900, so 0.96S = 48,000 and S = ₹50,000.

GST and sales tax. Price with GST = price × $\left(1 + \frac{r}{100}\right)$, and price before GST = price with GST × $\frac{100}{100 + r}$.

• An item of ₹2,400 with 12% GST costs 2,400 × 1.12 = ₹2,688.
• A bill of ₹5,900 that includes 18% GST has a base price of 5,900 × $\frac{100}{118}$ = ₹5,000, so the GST is ₹900.
• If the tax rate falls from $4\frac{1}{2}\%$ to 4%, a buyer saves 0.5% of the price: on ₹9,600 that is ₹48.

Income tax with slabs. Slabs work like commission slabs. With made-up slabs of nil on the first ₹3,00,000, 5% on the next ₹4,00,000 and 10% above ₹7,00,000, an income of ₹9,00,000 pays 0 + 20,000 + 20,000 = ₹40,000. That is about 4.44% of the income, even though the top rate is 10%. Real slabs change from year to year, so exam questions always state them.

Runs and other scores. "A batsman scores 150 runs, including 8 fours and 6 sixes. What % of his runs came from running between the wickets?" Boundaries = 32 + 36 = 68 runs, so running = 82 runs, which is 82 ÷ 150 × 100 = $54\frac{2}{3}\%$.

Attendance, absentees and the like. "On a day 12% of 250 workers were absent." That is 30 absent and 220 present. Any "what % of" question in daily life is one multiplication or division.

[SSC Tip] With a GST-inclusive price, never take r% of the final bill and subtract it. 18% of ₹5,900 is ₹1,062, but the GST in that bill is only ₹900, because it was charged on ₹5,000.

Solved Examples

Question 1

In an election between two candidates, the winner got 58% of the valid votes and won by 3,520 votes. How many valid votes were cast?

Answer. The loser got 42%, so the margin is 58 − 42 = 16% of the valid votes. 16% is 3,520, so 1% is 220 and the valid votes are 22,000.

Question 2

In an election, 15% of the voters on the list did not vote and 4% of the votes cast were invalid. Of the two candidates, the winner got 55% of the valid votes and the loser got 29,376 votes. How many voters were on the list?

Answer. The loser got 45% of the valid votes, so the valid votes = 29,376 ÷ 0.45 = 65,280. These are 96% of the votes cast, so the votes cast = 65,280 ÷ 0.96 = 68,000. Those are 85% of the list, so the list had 68,000 ÷ 0.85 = 80,000 voters. In one line: 29,376 ÷ (0.85 × 0.96 × 0.45).

Question 3

In an election, 12% of the voters on the list did not vote and 84 votes were found invalid. There were two candidates. The winner got the votes of 46% of all the voters on the list and got 552 votes more than his rival. Find the number of voters on the list.

Answer. Say N voters are on the list. Votes cast = 0.88N, so valid votes = 0.88N − 84. The winner got 0.46N, so the loser got 0.88N − 84 − 0.46N = 0.42N − 84. The gap is 0.46N − (0.42N − 84) = 0.04N + 84 = 552. So 0.04N = 468 and N = 11,700. Check: the winner got 5,382 and the loser 4,830, a difference of 552.

Question 4

Three candidates contested an election. 10% of the votes polled were invalid. Of the valid votes, A got 45%, B got 35% and C got the rest. A beat C by 9,000 votes. How many votes were polled?

Answer. C got 100 − 45 − 35 = 20% of the valid votes. A − C = 25% of the valid votes = 9,000, so the valid votes are 36,000. They are 90% of the votes polled, so the votes polled = 36,000 ÷ 0.9 = 40,000.

Question 5

To win an election, a candidate needs at least 40% of the votes polled. A candidate got 32% of the votes and fell short of the winning mark by 2,400 votes. How many votes were polled? (a) 6,000 (b) 7,500 (c) 24,000 (d) 30,000

Answer. She was 40 − 32 = 8% short, and that 8% is 2,400 votes. So 1% is 300 and the votes polled are 30,000. Dividing 2,400 by 40% instead of by the 8% gap gives option (a). The answer is (d) 30,000.

Question 6

[Tier 2] In an election between two candidates, A won by a margin of 12% of the votes polled. If 1,800 voters had voted for B instead of A, B would have won by 8% of the votes polled. Find the total number of votes polled.

Answer. Each voter who switches takes one vote from A and gives one to B, so the gap moves by 2 × 1,800 = 3,600 votes. The gap goes from A ahead by 12% to B ahead by 8%, a swing of 20% of the votes polled. So 20% is 3,600 and the votes polled are 18,000. Check: A got 10,080 and B 7,920 (gap 2,160, which is 12%); after the switch A has 8,280 and B 9,720 (gap 1,440, which is 8%).

Question 7

A student needs 36% of the maximum marks to pass. He scores 247 marks and fails by 23 marks. Find the maximum marks.

Answer. Pass marks = 247 + 23 = 270. This is 36% of the maximum, so the maximum = 270 ÷ 0.36 = 750.

Question 8

In an exam, Ajay scores 28% and fails by 42 marks. Vijay scores 43% and gets 48 marks more than the pass mark. Find the maximum marks and the pass percentage.

Answer. The pass mark sits between them: Ajay is 42 below it and Vijay 48 above it. So the 15% gap between them is 42 + 48 = 90 marks, which makes 1% = 6 marks and the maximum 600. Pass mark = 28% of 600 + 42 = 168 + 42 = 210, and 210 ÷ 600 is 35%. Check: 43% of 600 = 258 = 210 + 48.

Question 9

The pass mark in an exam is 33%. Neha scored 240 marks and failed by 24 marks. The board then gave every student grace marks equal to 4% of the maximum marks. Did Neha pass, and by how many marks?

Answer. Pass mark = 240 + 24 = 264, which is 33% of the maximum, so the maximum is 264 ÷ 0.33 = 800. Grace marks = 4% of 800 = 32. Her new score is 272, so she passes by 8 marks.

Question 10

A student took three papers with maximum marks of 100, 150 and 250. She scored 68%, 72% and 80% in them. What is her overall percentage?

Answer. Her marks are 68, 108 and 200, a total of 376 out of 500. Overall = 376 ÷ 500 × 100 = 75.2%. The plain average of 68, 72 and 80 is $73\frac{1}{3}\%$, which is wrong here because the papers carry different marks.

Question 11

In an exam, 32% of the students failed in Mathematics, 26% failed in Science and 12% failed in both. If 1,080 students passed in both subjects, how many students took the exam?

Answer. Failed in at least one subject = 32 + 26 − 12 = 46%. So 54% passed in both, and 54% is 1,080. That makes 1% = 20, so 2,000 students took the exam.

Question 12

In a class, 70% of the students passed in English, 60% passed in Hindi and 45% passed in both. What percentage failed in both, and what percentage failed in exactly one subject?

Answer. Passed in at least one = 70 + 60 − 45 = 85%, so 15% failed in both. Passed only in English = 70 − 45 = 25% (they failed Hindi), and passed only in Hindi = 60 − 45 = 15% (they failed English). So 40% failed in exactly one subject. Check: 45 + 25 + 15 + 15 = 100.

Question 13

Rahul spends 25% of his salary on rent, 20% of the remaining on food and 15% of what is left after that on travel. He saves the rest, which is ₹15,300. Find his salary.

Answer. Each "of the remaining" step multiplies what is left. After rent: 0.75. After food: 0.75 × 0.8 = 0.6. After travel: 0.6 × 0.85 = 0.51. So 51% of his salary is ₹15,300, and the salary is 15,300 ÷ 0.51 = ₹30,000.

Question 14

Mohan spends 30% of his salary on rent and 20% of his salary on food. Of the rest, he spends 40% on travel and saves the remaining ₹10,800. Find his salary.

Answer. Rent and food are both "of salary", so together they take 50% and leave 50%. He spends 40% of that and saves 60% of it, and 60% of 50% is 30% of the salary. So 30% is ₹10,800 and the salary is ₹36,000. Treating the food as 20% of the remainder would be a mistake, because the question says 20% of his salary.

Question 15

A man gives 20% of his money to his wife, 30% of the rest to his son and 25% of what is then left to a charity. He is left with ₹25,200. How much money did he have at first?

Answer. What is left = 0.8 × 0.7 × 0.75 = 0.42 of his money. So 42% is ₹25,200, and he had 25,200 ÷ 0.42 = ₹60,000. (His wife got ₹12,000, his son ₹14,400 and the charity ₹8,400.)

Question 16

A family's monthly budget is split like this: food 35%, rent 20%, education 15%, transport 8%, others 10% and savings 12%. (i) What is the central angle for education in a pie chart? (ii) If the monthly income is ₹45,000, how much more is spent on food than is saved?

Answer. (i) 1% is 3.6°, so education gets 15 × 3.6 = 54°. (ii) Food is 35% and savings are 12%, a difference of 23% of the income. 23% of ₹45,000 = ₹10,350.

Question 17

School A sent 480 students for an exam and 75% passed. School B sent 320 and 60% passed. Find the overall pass percentage.

Answer. Passed: 75% of 480 = 360 and 60% of 320 = 192, a total of 552 out of 800. Overall = 552 ÷ 800 × 100 = 69%. The plain average of 75 and 60 is 67.5%, but School A is bigger, so the true figure is pulled towards 75.

Question 18

A town has 64,000 people, and 58% of them are men. 45% of the men and 30% of the women are literate. Find the literacy rate of the town and the number of literate people.

Answer. As a weighted average: 0.58 × 45 + 0.42 × 30 = 26.1 + 12.6 = 38.7%. Literate people = 38.7% of 64,000 = 24,768. (There are 37,120 men, of whom 16,704 are literate, and 26,880 women, of whom 8,064 are literate.)

Question 19

A 60-litre mixture of milk and water has 15% water. How much water must be added so that water makes up 25% of the mixture?

Answer. The milk does not change: 85% of 60 = 51 L. In the new mixture milk is 75%, so the new total is 51 ÷ 0.75 = 68 L. Water to add = 68 − 60 = 8 litres.

Question 20

Fresh grapes contain 85% water and raisins contain 10% water. How many kg of raisins can be made from 36 kg of fresh grapes?

Answer. Only water dries out; the solid part stays. Solid in the fresh grapes = 15% of 36 = 5.4 kg. In raisins this solid is 90% of the weight, so the raisins weigh 5.4 ÷ 0.9 = 6 kg.

Question 21

30 litres of a 20% acid mixture is mixed with 20 litres of a 45% acid mixture. What is the acid percentage in the result?

Answer. Acid: 20% of 30 = 6 L and 45% of 20 = 9 L, so 15 L of acid in 50 L. That is 15 ÷ 50 × 100 = 30%.

Question 22

A salesman gets 5% commission on sales up to ₹20,000 and 4% on sales above ₹20,000. After deducting his commission, he sent ₹76,600 to his company. Find his total sales.

Answer. Say the sales are S (more than ₹20,000). Commission = 5% of 20,000 + 4% of (S − 20,000) = 1,000 + 0.04S − 800 = 0.04S + 200. He sent S − (0.04S + 200) = 0.96S − 200 = 76,600. So 0.96S = 76,800 and S = ₹80,000. Check: commission = 1,000 + 2,400 = ₹3,400, and 80,000 − 3,400 = 76,600.