If a, b and c are three positive integers such that a and b are in the ratio 3:4 while b and c are in the ratio 2:1, then minimum integer value of a + b + c is _________Correct answer is '9'. Can you explain this answer?

Question

Answer

Given Information:

a and b are in the ratio 3:4

b and c are in the ratio 2:1

Solution:

To find the minimum integer value of a, b, and c, we need to determine the smallest possible values for a, b, and c that satisfy the given ratios.

Finding the Ratio between a, b, and c:

Given that a and b are in the ratio 3:4, we can assume that a is a multiple of 3 and b is a multiple of 4. Let's represent this as:
a = 3x
b = 4x

Similarly, since b and c are in the ratio 2:1, we can assume that b is a multiple of 2 and c is a multiple of 1. Let's represent this as:
b = 2y
c = 1y

Combining the Ratios:

Now, let's combine the ratios of a and b, and b and c to establish a relation between a, b, and c.

We have:
a:b = 3:4
b:c = 2:1

Substituting the values of a and b from the previous step, we get:
(3x):(4x) = 3:4
(4x):(2y) = 2:1

Simplifying the ratios, we get:
3x/4x = 3/4
4x/2y = 2/1

Cross-multiplying and simplifying, we get:
3x = 4(3)
4x = 2(2y)

Solving these equations, we find:
x = 4
y = 4

Calculating the Minimum Values:

Now that we have the values of x and y, we can substitute them back into the equations of a, b, and c to find their minimum values.

a = 3x = 3(4) = 12
b = 4x = 4(4) = 16
c = 1y = 1(4) = 4

Therefore, the minimum integer values of a, b, and c that satisfy the given ratios are:
a = 12
b = 16
c = 4

Conclusion:

The minimum integer value of a * b * c is 12 * 16 * 4 = 768.

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