CMAT 2026: Arithmetic
Arithmetic Quick Review
Successive Discounts:
Equivalent Discount = \(x + y – \frac{xy}{100}\).
For three discounts, apply to first two, then result with the third.
Equivalent Discount = \(x + y – \frac{xy}{100}\).
For three discounts, apply to first two, then result with the third.
Mixtures (Ratio Method):
Total Quantity / Total Ratio Parts = Value per part.
Identify what remains constant (usually Milk or Initial Solvent).
Total Quantity / Total Ratio Parts = Value per part.
Identify what remains constant (usually Milk or Initial Solvent).
Time and Work:
Work Done = Rate \(\times\) Time.
Total Work is assumed as 1 or LCM of individual times.
Work Done = Rate \(\times\) Time.
Total Work is assumed as 1 or LCM of individual times.
85 litres of a mixture contain milk and water in the ratio 27:7. How much more water is required to be added in the mixture, so that the resulting mixture contains milk and water in the ratio 3:1?
Correct Option: 5 litres
Rationale:
1. Original: 85L, Ratio 27:7. Total parts = 34.
2. Milk = \(\frac{27}{34} \times 85 = 67.5\)L. Water = \(\frac{7}{34} \times 85 = 17.5\)L.
3. New Ratio = 3:1. Milk remains constant (67.5L).
4. \(\frac{67.5}{17.5 + x} = \frac{3}{1} \implies 67.5 = 3(17.5 + x)\).
5. \(22.5 = 17.5 + x \implies x = 5\) litres.
Difficulty: Moderate
Rationale:
1. Original: 85L, Ratio 27:7. Total parts = 34.
2. Milk = \(\frac{27}{34} \times 85 = 67.5\)L. Water = \(\frac{7}{34} \times 85 = 17.5\)L.
3. New Ratio = 3:1. Milk remains constant (67.5L).
4. \(\frac{67.5}{17.5 + x} = \frac{3}{1} \implies 67.5 = 3(17.5 + x)\).
5. \(22.5 = 17.5 + x \implies x = 5\) litres.
Difficulty: Moderate
Sudhir purchased a chair with three successive discounts of 20%, 12.5% and 5%. The equivalent single discount is:
Correct Option: 33.5%
Rationale:
Let MP = 100.
1. After 20% disc: \(100 – 20 = 80\).
2. After 12.5% (\(1/8\)) disc: \(80 – 10 = 70\).
3. After 5% disc: \(70 – (5\% \text{ of } 70) = 70 – 3.5 = 66.5\).
Total Discount = \(100 – 66.5 = 33.5\).
Why other options wrong: Summing percentages directly (37.5%) is a common trap.
Difficulty: Moderate
Rationale:
Let MP = 100.
1. After 20% disc: \(100 – 20 = 80\).
2. After 12.5% (\(1/8\)) disc: \(80 – 10 = 70\).
3. After 5% disc: \(70 – (5\% \text{ of } 70) = 70 – 3.5 = 66.5\).
Total Discount = \(100 – 66.5 = 33.5\).
Why other options wrong: Summing percentages directly (37.5%) is a common trap.
Difficulty: Moderate
Even after reducing the marked price of an item, by Rs. 32, a shopkeeper makes a profit of 15%. If the cost price is Rs. 320, what percentage of profit would he have made, if he had sold the item at the marked price?
Correct Option: 25%
Rationale:
CP = 320. Profit = 15% of 320 = 48.
SP = \(320 + 48 = 368\).
MP = \(SP + \text{Discount} = 368 + 32 = 400\).
New Profit at MP = \(400 – 320 = 80\).
Profit % = \(\frac{80}{320} \times 100 = 25\%\).
Difficulty: Easy
Rationale:
CP = 320. Profit = 15% of 320 = 48.
SP = \(320 + 48 = 368\).
MP = \(SP + \text{Discount} = 368 + 32 = 400\).
New Profit at MP = \(400 – 320 = 80\).
Profit % = \(\frac{80}{320} \times 100 = 25\%\).
Difficulty: Easy
If height and radius of right circular cylinder are equal and its volume is 176/7 cm\(^3\) then the diameter of the cylinder, is:
Correct Option: 1 (4 cm)
Rationale: Given \(h = r\). Volume \(V = \pi r^2 h = \pi r^3\).
\(\frac{176}{7} = \frac{22}{7} \times r^3 \implies 176 = 22r^3 \implies r^3 = 8\).
\(r = 2\) cm. Diameter = \(2r = 4\) cm.
Difficulty: Easy
Rationale: Given \(h = r\). Volume \(V = \pi r^2 h = \pi r^3\).
\(\frac{176}{7} = \frac{22}{7} \times r^3 \implies 176 = 22r^3 \implies r^3 = 8\).
\(r = 2\) cm. Diameter = \(2r = 4\) cm.
Difficulty: Easy
Consider the following statements about numbers.
A. There exists a smallest natural number.
B. There exists a largest natural number.
C. Every rational number is also a real number.
D. Between two consecutive natural numbers, there is always a natural number.
Choose the correct answer from the options given below:
A. There exists a smallest natural number.
B. There exists a largest natural number.
C. Every rational number is also a real number.
D. Between two consecutive natural numbers, there is always a natural number.
Choose the correct answer from the options given below:
Correct Option: A and C only
Rationale:
A: True (Smallest natural number is 1).
B: False (Natural numbers are infinite).
C: True (Real numbers include Rationals and Irrationals).
D: False (No natural number exists between 5 and 6).
Difficulty: Easy
Rationale:
A: True (Smallest natural number is 1).
B: False (Natural numbers are infinite).
C: True (Real numbers include Rationals and Irrationals).
D: False (No natural number exists between 5 and 6).
Difficulty: Easy
Raman can do a piece of work in 16 days. Satish can do the same work in 8 days while Ashok can do it in 32 days. All of them worked together when they started, but Satish left after 2 days. Raman left 3 days before the completion of work. How long did it take to complete the entire work?
Correct Option: 10 days
Rationale: Let total time be \(T\). Rates: Raman \(1/16\), Satish \(1/8\), Ashok \(1/32\).
Work Equation: \(\frac{2}{8} + \frac{T-3}{16} + \frac{T}{32} = 1\).
Multiply by 32: \(8 + 2(T-3) + T = 32\).
\(8 + 2T – 6 + T = 32 \implies 3T + 2 = 32 \implies 3T = 30\).
\(T = 10\) days.
Difficulty: Moderate
Rationale: Let total time be \(T\). Rates: Raman \(1/16\), Satish \(1/8\), Ashok \(1/32\).
Work Equation: \(\frac{2}{8} + \frac{T-3}{16} + \frac{T}{32} = 1\).
Multiply by 32: \(8 + 2(T-3) + T = 32\).
\(8 + 2T – 6 + T = 32 \implies 3T + 2 = 32 \implies 3T = 30\).
\(T = 10\) days.
Difficulty: Moderate
