Arithmetic — CUET UG General Test

Arithmetic tests the same core tools as Numerical Ability — percentage, ratio, averages — but in their applied, business-flavoured form: a percentage OF a group rather than a number in isolation, a profit or savings percentage rather than a bare increase, and an average across two combined groups rather than one. The calculations stay simple; the framing is what needs careful reading.

1. Percentage of a group

"What percentage of a group has a given property" is computed exactly like any percentage — the property's count divided by the group's total count, times 100 — but the framing as a "share of a whole" is what distinguishes it from a bare percentage-of-a-number question. In a class of 40 students with 15 girls, the girls' share is $\frac{15}{40}\times 100=37.5\%$.

A closely related question asks "X is what percentage of Y," which is solved the same way — divide X by Y, multiply by 100 — regardless of which quantity is the "part" and which is the "whole."

2. Percentage increase and profit percent

Percentage increase compares a NEW value to an OLD value: the change divided by the OLD value, times 100. Profit percent is the same calculation applied specifically to buying and selling — profit (selling price minus cost price) divided by the COST price, never the selling price, times 100.

$\text{Profit \%}=\frac{\text{Selling Price}-\text{Cost Price}}{\text{Cost Price}}\times 100$

A product bought for ₹250 and sold for ₹310 earns a profit of ₹60, and $\frac{60}{250}\times 100=24\%$ profit — dividing by the selling price of ₹310 instead would give a different, incorrect percentage.

3. Savings percentage

Savings percentage follows the identical structure: (income minus expenditure) divided by INCOME, times 100 — not divided by expenditure. A person earning ₹25,000 and spending ₹18,000 saves ₹7,000, giving a savings percentage of $\frac{7000}{25000}\times 100=28\%$.

4. Reverse percentage on a group or total

A reverse-percentage question at the group level gives one part's percentage share along with a numeric DIFFERENCE between two parts, and asks for the total — the same "divide by the multiplier" logic as single-number reverse percentage, applied to a percentage-point gap instead. If a winning candidate gets 60% of an election's total votes and wins by 4,800 votes over the only other candidate, that 4,800-vote margin represents $60\%-40\%=20\%$ of the total votes, so the total is $4800÷0.20=24,000$.

5. Ratio comparison

Comparing two ratios to see which is larger is done by converting each to a decimal (or a common denominator) and comparing directly — there is no shortcut around actually computing both values. $\frac{3}{4}=0.75$ and $\frac{5}{7}\approx 0.714$, so the ratio 3:4 is larger than 5:7, even though 5:7 might look larger at a glance because both its terms are bigger numbers.

6. Three-part ratios and difference of shares

When a total is split in a three-part ratio and the question gives the numeric DIFFERENCE between two of the three shares, that difference corresponds to the difference between their respective ratio terms. Dividing the numeric difference by the term-difference recovers the value of one "part," which then scales to the full total.

If a sum is split among X, Y, Z in the ratio 3:5:8 and Z receives ₹900 more than X, that ₹900 spans $8-3=5$ parts, so one part is ₹180 and the full total (16 parts) is ₹2,880.

7. Average: recovering the sum

Since average equals sum divided by count, the sum is recovered by multiplying the average back by the count — the single most useful reverse-direction move in this whole topic, used identically whether the question then asks for a removed value, an added value, or a group total.

8. Average with an added value

When a NEW value joins a group and the average changes, the new value is found by comparing the OLD total sum to the NEW total sum — the difference between the two sums is exactly the value that was added. If 6 numbers average 35 and a 7th number changes the average to 36, the old sum is $6\times 35=210$ and the new sum is $7\times 36=252$, so the added number is $252-210=42$.

9. Combined average of two groups

The average of two combined groups is NOT the simple average of their two individual averages — it is the combined SUM of both groups divided by their combined COUNT, which only equals the simple average when both groups have equal size.

Class A (30 students, average 70) combined with Class B (20 students, average 80) gives a combined average of $\frac{30\times 70+20\times 80}{50}=\frac{3700}{50}=74$ — closer to 70 than to the midpoint 75, because the larger group pulls the combined average toward its own value.

Worked Examples

Example 1. In a survey of 80 people, 24 preferred product A. What percentage of the surveyed people preferred product A?

$\frac{24}{80}\times 100=30\%$.

Answer: 30%.

Example 2. 21 is what percentage of 60?

$\frac{21}{60}\times 100=35\%$.

Answer: 35%.

Example 3. A trader buys an item for ₹480 and sells it for ₹600. Find the profit percentage.

Profit $=600-480=120$. Profit % $=\frac{120}{480}\times 100=25\%$ (dividing by the cost price, ₹480, not the selling price).

Answer: 25%.

Example 4. A family earns ₹40,000 a month and spends ₹28,000. Find their savings percentage.

Savings $=40000-28000=12000$. Savings % $=\frac{12000}{40000}\times 100=30\%$.

Answer: 30%.

Example 5. A shopkeeper's revenue increased from ₹4,500 to ₹5,850 in a month. Find the percentage increase.

$\frac{5850-4500}{4500}\times 100=\frac{1350}{4500}\times 100=30\%$.

Answer: 30%.

Example 6. In an election with only two candidates, the winner received 55% of the total votes and won by 2,000 votes. Find the total number of votes cast.

The winning margin represents $55\%-45\%=10\%$ of the total. Total $=2000÷0.10=20,000$.

Answer: 20,000 votes.

Example 7. Which ratio is larger: 4:5 or 7:9?

$\frac{4}{5}=0.8$ and $\frac{7}{9}\approx 0.778$. Since $0.8>0.778$, the ratio 4:5 is larger.

Answer: 4:5 is larger.

Example 8. A sum of money is divided among P, Q and R in the ratio 2:3:7. If R receives ₹1,000 more than P, find the total sum.

R and P's ratio terms differ by $7-2=5$ parts, corresponding to ₹1,000, so one part $=1000÷5=₹200$. Total parts $=2+3+7=12$, so the total sum is $200\times 12=₹2,400$.

Answer: ₹2,400.

Example 9. The average of 8 numbers is 42. If one more number is included, the new average becomes 44 for all 9 numbers. Find the included number.

Old sum $=8\times 42=336$. New sum $=9\times 44=396$. Included number $=396-336=60$.

Answer: 60.

Example 10. A group of 25 students has an average score of 60, and another group of 15 students has an average score of 80. Find the combined average score of all 40 students.

Combined sum $=25\times 60+15\times 80=1500+1200=2700$. Combined average $=\frac{2700}{40}=67.5$.

Answer: 67.5.

Summary

Percentage OF a group and "X is what percentage of Y" both compute the same way — part divided by whole, times 100 — regardless of which framing the question uses.

Profit percent and savings percentage both divide by the ORIGINAL base value (cost price for profit, income for savings), never by the other quantity. Reverse percentage at the group level converts a numeric difference between two shares into a percentage-point gap, then divides by that gap to recover the total.

Comparing two ratios requires actually converting both to decimals or a common denominator — there is no shortcut from the raw numbers alone. A three-part ratio's difference-of-shares question recovers one "part's" value from the numeric gap between two ratio terms, then scales to the full total.

Recovering a sum from an average (Sum = Average x Count) underlies every average question in this chapter — an added value is the difference between the new and old sums, and a combined average of two groups is their combined sum divided by their combined count, never the simple average of the two group averages.