Arithmetic

Ratio, Proportion & Variation

Medium

The number of coins collected per week by two coin-collectors $A$ and $B$ are In the ratio $3:4$ . If the total number of coins collected by $A$ in $5$ weeks is a multiple of $7$ , and the total number of coins collected by $B$ in $3$ weeks is a multiple of $24$ , then the minimum possible number of coins collected by $A$ in one week is

Entered answer:

✅ Correct Answer: 42

We break down what we know:

Two coin collectors A and B collect coins each week. Their collection rates are in the ratio 3:4. A's total collection in 5 weeks is a multiple of 7. B's total collection in 3 weeks is a multiple of 24.

When we say A and B collect coins in the ratio 3:4, this means:

If A collects 3 coins per week, then B collects 4 coins per week. If A collects 6 coins per week, then B collects 8 coins per week. And so on...

The key insight: We can represent their weekly collections as:

A collects 3x coins per week

B collects 4x coins per week

where x is some positive whole number that maintains the 3:4 ratio.

Now we calculate their total collections over multiple weeks:

A's collection in 5 weeks: $5 \times 3x = 15x$ coins

B's collection in 3 weeks: $3 \times 4x = 12x$ coins

A number is a multiple of 7 if it can be written as $7 \times$ (some whole number).

Examples: 7, 14, 21, 28, 35...

A number is a multiple of 24 if it can be written as $24 \times$ (some whole number).

Examples: 24, 48, 72, 96...

Condition 1: $15x$ is a multiple of 7

This means $15x = 7k$ for some whole number k.

Since 15 and 7 share no common factors (they are coprime), x itself must be a multiple of 7.

Condition 2: $12x$ is a multiple of 24

This means $12x = 24m$ for some whole number m.

Dividing both sides by 12: $x = 2m$

This tells us x must be even (divisible by 2).

We need x to satisfy both conditions:

x must be a multiple of 7

x must be even

The smallest positive number that is both a multiple of 7 and even is 14.

We verify:

Is 14 a multiple of 7? Yes: $14 = 7 \times 2$

Is 14 even? Yes: $14 = 2 \times 7$

With $x = 14$ :

A collects $3x = 3 \times 14 = 42$ coins per week

We check:

A in 5 weeks: $42 \times 5 = 210$ coins ( $210 \div 7 = 30$ , so it's a multiple of 7)

B in 3 weeks: $56 \times 3 = 168$ coins ( $168 \div 24 = 7$ , so it's a multiple of 24)

Therefore, the minimum possible number of coins collected by A in one week is 42.

CAT 2023 Slot 1

The salaries of three friends Sita, Gita and Mita are initially in the ratio $5 : 6 : 7$ , respectively. In the first year, they get salary hikes of $20\%$ , $25\%$ and $20\%$ , respectively. In the second year, Sita and Mita get salary hikes of $40\%$ and $25\%$ , respectively, and the salary of Gita becomes equal to the mean salary of the three friends. The salary hike of Gita in the second year is

CAT 2017 Slot 1

Suppose, $C1, C2, C3, C4,$ and $C5$ are five companies. The profits made by $C1, C2,$ and $C3$ are in the ratio $9: 10: 8$ while the profits made by $C2, C4,$ and $C5$ are in the ratio $18:19:20$ . If $C5$ has made a profit of Rs $19$ crore more than $C1$ , then the total profit (in Rs) made by all five companies is

2025 Slot 1

The ratio of the number of students in the morning shift and afternoon shift of a school was 13 : 9. After 21 students moved from the morning shift to the afternoon shift, this ratio became 19 : 14. Next, some new students joined the morning and afternoon shifts in the ratio 3 : 8 and then the ratio of the number of students in the morning shift and the afternoon shift became 5 : 4. The number of new students who joined is

CAT 2018 Slot 1

Raju and Lalitha originally had marbles in the ratio $4:9$ . Then Lalitha gave some of her marbles to Raju. As a result, the ratio of the number of marbles with Raju to that with Lalitha became $5:6$ . What fraction of her original number of marbles was given by Lalitha to Raju?

CAT 2020 Slot 1

An alloy is prepared by mixing three metals $A, B$ and $C$ in the proportion $3 : 4 :7$ by volume. Weights of the same volume of the metals $A, B$ and $C$ are in the ratio $5:2: 6.$ In $130$ kg of the alloy, the weight, in kg, of the metal $C$ is

2025 Slot 2

The ratio of expenditures of Lakshmi and Meenakshi is $2:3$ , and the ratio of income of Lakshmi to expenditure of Meenakshi is $6:7$ . If excess of income over expenditure is saved by Lakshmi and Meenakshi, and the ratio of their savings is $4:9$ , then the ratio of their incomes is

Solution

Related questions:

Keyboard Shortcuts

CAT Past Year Questions (Topic-Wise)