CAT Percentages Bible Top 10 Must Solve Percentages Questions covering 80% of Syllabus – Page 3
All 40 Percentages questions from CAT papers. Tap Show answer, or open the detailed solution (free login). Page 3 of 10.
A glass is filled with milk. Two-thirds of its content is poured out and replaced with water.
If this process of pouring out two-thirds the content and replacing with water is repeated
three more times, then the final ratio of milk to water in the glass is:
If this process of pouring out two-thirds the content and replacing with water is repeated
three more times, then the final ratio of milk to water in the glass is:
Answer: D. 1 : 80
Solution — first steps
Step 1: Concept of Dilution and Replacement
When a fraction of a liquid is replaced repeatedly, the amount of the original liquid
left after \( n \) repetitions is given by:
\[
\text{Remaining Fraction of Original Liquid} = \left( 1 - \text{Fraction Replaced} \right)^n
\]
Step 2: Apply the formula
Fraction replaced = \( \frac{2}{3} \), so the remaining fraction = \( 1 - \frac{2}{3} = \frac{1}{3} \).
Number of repetitions, \( n = 4 \).
\[
\text{Fraction of Milk Remaining} = \left( \frac{1}{3} \right)^4 = \frac{1}{81}
\]
When a fraction of a liquid is replaced repeatedly, the amount of the original liquid
left after \( n \) repetitions is given by:
\[
\text{Remaining Fraction of Original Liquid} = \left( 1 - \text{Fraction Replaced} \right)^n
\]
Step 2: Apply the formula
Fraction replaced = \( \frac{2}{3} \), so the remaining fraction = \( 1 - \frac{2}{3} = \frac{1}{3} \).
Number of repetitions, \( n = 4 \).
\[
\text{Fraction of Milk Remaining} = \left( \frac{1}{3} \right)^4 = \frac{1}{81}
\]
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
In September, the incomes of Kamal, Amal and Vimal are in the ratio 8 : 6 : 5. They rent a house together, and Kamal pays 15%, Amal pays 12% and Vimal pays 18% of their respective incomes to cover the total house rent in that month. In October, the house rent remains unchanged while their incomes increase by 10%, 12% and 15%, respectively. In October, the percentage of their total income that will be paid as house rent, is nearest to
Answer: C. 13.26
Solution — first steps
Step 1. Represent September incomes as 8u,6u,5u.
Step 2. September rent is 15% of 8u plus 12% of 6u plus 18% of 5u =1.2u+0.72u+0.9u=2.82u.
Step 2. September rent is 15% of 8u plus 12% of 6u plus 18% of 5u =1.2u+0.72u+0.9u=2.82u.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
An amount of Rs 10000 is deposited in bank A for a certain number of years at a simple interest of 5% per annum. On maturity, the total amount received is deposited in bank B for another 5 years at a simple interest of 6% per annum. If the interests received from bank A and bank B are in the ratio 10 : 13, then the investment period, in years, in bank A is
Answer: C. 6
Solution — first steps
Step 1. Let the term at bank A be n years. Its simple interest is 10000×0.05n=500n.
Step 2. The maturity deposited in B is 10000+500n.
Step 2. The maturity deposited in B is 10000+500n.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
A vessel contained a certain amount of a solution of acid and water. When 2 litres of water was added to it, the new solution had 50% acid concentration. When 15 litres of acid was further added to this new solution, the final solution had 80% acid concentration. The ratio of water and acid in the original solution was
Answer: C. 3 : 5
Solution — first steps
Step 1: After adding 2 litres of water, the concentration of acid becomes 50%.
\( \frac{A}{A + W + 2} = \frac{1}{2} \)
\( \Rightarrow 2A = A + W + 2 \Rightarrow A = W + 2 \quad \text{(Equation 1)} \)
Step 2: Then, 15 litres of acid is added and acid becomes 80% of the solution:
\( \frac{A + 15}{A + W + 17} = \frac{4}{5} \)
\( \Rightarrow 5(A + 15) = 4(A + W + 17) \)
\( \Rightarrow 5A + 75 = 4A + 4W + 68 \)
\( \Rightarrow A = 4W - 7 \quad \text{(Equation 2)} \)
\( \frac{A}{A + W + 2} = \frac{1}{2} \)
\( \Rightarrow 2A = A + W + 2 \Rightarrow A = W + 2 \quad \text{(Equation 1)} \)
Step 2: Then, 15 litres of acid is added and acid becomes 80% of the solution:
\( \frac{A + 15}{A + W + 17} = \frac{4}{5} \)
\( \Rightarrow 5(A + 15) = 4(A + W + 17) \)
\( \Rightarrow 5A + 75 = 4A + 4W + 68 \)
\( \Rightarrow A = 4W - 7 \quad \text{(Equation 2)} \)
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
Bina incurs 19% loss when she sells a product at Rs. 4860 to Shyam, who in turn sells this product to Hari. If Bina would have sold this product to Shyam at the purchase price of Hari, she would have obtained 17% profit. Then, the profit, in rupees, made by Shyam is
Answer: 2160
Solution — first steps
Step 1. Bina sells at 19% loss for 4860, so her cost is 4860/0.81=6000.
Step 2. If Bina had sold at Hari's purchase price, her profit would be 17%, so Hari paid 1.17×6000=7020.
Step 2. If Bina had sold at Hari's purchase price, her profit would be 17%, so Hari paid 1.17×6000=7020.
The full step-by-step working opens with a free login.
For students who have joined a Cetking program.
Last day solving checklist
- Name the unknown. Include its unit and state whether it is a count, amount, rate or parameter.
- Form the relationships. Translate each given condition into an equation or restriction.
- Choose the method. Use the formula, substitution, elimination, a sign chart or a bound appropriate to this concept.
- Check restrictions. Check integer counts, signs, nonzero denominators, domains and all original conditions.
- Verify the answer. Substitute into the original question; for an extreme, show a feasible example reaches it.
All 16 repeating CAT PYQs and how much each adds
| CAT PYQ | Repeating CAT PYQ | Questions | Representative | Cumulative questions | Cumulative share |
|---|---|---|---|---|---|
| 1 | Repeated fractional replacement | 6 | CAT 2021 S2 Q4 | 6 | 10.71% |
| 2 | Compound growth factors and equivalent simple interest | 6 | CAT 2021 S1 Q10 | 12 | 21.43% |
| 3 | Cost marked price discount and free item multipliers | 5 | CAT 2024 S1 Q21 | 17 | 30.36% |
| 4 | Successive percentage changes with unknown rates | 5 | CAT 2021 S2 Q18 | 22 | 39.29% |
| 5 | Weighted subgroup payroll or investment growth | 5 | CAT 2023 S2 Q8 | 27 | 48.21% |
| 6 | Percentages of groups and remaining shares | 5 | CAT 2021 S2 Q7 | 32 | 57.14% |
| 7 | Weighted sale revenue and overall profit | 4 | CAT 2021 S2 Q22 | 36 | 64.29% |
| 8 | Profit shares and successive trading transactions | 4 | CAT 2021 S2 Q8 | 40 | 71.43% |
| 9 | Percentage stocks constrained to integer counts | 3 | CAT 2023 S3 Q14 | 43 | 76.79% |
| 10 | Component percentages after adding mixtures | 3 | CAT 2021 S3 Q1 | 46 | 82.14% |
| 11 | Compound balances after repayments | 2 | CAT 2023 S2 Q12 | 48 | 85.71% |
| 12 | Workforce and efficiency percentages | 2 | CAT 2022 S3 Q16 | 50 | 89.29% |
| 13 | Simple interest rates and reinvestment | 2 | CAT 2022 S1 Q16 | 52 | 92.86% |
| 14 | Percentage changes in speed or angle | 2 | CAT 2025 S3 Q16 | 54 | 96.43% |
| 15 | Unit price percentage and quantity ratio | 1 | CAT 2021 S1 Q3 | 55 | 98.21% |
| 16 | Reciprocal day growth factors | 1 | CAT 2022 S2 Q7 | 56 | 100% |
Based on CAT Actual Quant Papers.