Understanding the Ratio Problem with Tokens

The problem asks us to find the combined ratio of red(A), yellow(B), and green(C) tokens in a bag, given the individual ratios A:B and B:C. We are given:

• The ratio of red tokens (A) to yellow tokens (B) is A : B ∷ 3 : 8.
• The ratio of yellow tokens (B) to green tokens (C) is B : C ∷ 6 : 13.

Our goal is to find the composite ratio A : B : C.

Method for Combining Ratios

To combine two ratios that share a common element, we need to make the value corresponding to the common element the same in both ratios. In this case, the common element is the yellow tokens (B). We need to find a common value for B in both the A:B ratio (where B is 8) and the B:C ratio (where B is 6).

The easiest way to do this is to find the least common multiple (LCM) of the two values for B, which are 8 and 6.

Let's find the LCM of 8 and 6:

• Prime factorization of 8 is \(2 \times 2 \times 2 = 2^3\).
• Prime factorization of 6 is \(2 \times 3\).
• The LCM is found by taking the highest power of all prime factors present in either number: \(2^3 \times 3 = 8 \times 3 = 24\).

So, the LCM of 8 and 6 is 24. We need to adjust both ratios so that the value for B is 24.

Adjusting the Ratios

Adjusting the A : B ratio (3 : 8):

To change the '8' in A:B to '24', we need to multiply 8 by 3 (\(8 \times 3 = 24\)). To keep the ratio equivalent, we must multiply both parts of the ratio by 3:

\( (3 \times 3) : (8 \times 3) = 9 : 24 \)

So, the adjusted A : B ratio is 9 : 24.

Adjusting the B : C ratio (6 : 13):

To change the '6' in B:C to '24', we need to multiply 6 by 4 (\(6 \times 4 = 24\)). To keep the ratio equivalent, we must multiply both parts of the ratio by 4:

\( (6 \times 4) : (13 \times 4) = 24 : 52 \)

So, the adjusted B : C ratio is 24 : 52.

Combining the Adjusted Ratios

Now we have the adjusted ratios where the value for B is the same:

• A : B = 9 : 24
• B : C = 24 : 52

Since the value corresponding to B is now 24 in both ratios, we can combine them directly to find the ratio A : B : C.

The combined ratio A : B : C is 9 : 24 : 52.

Comparing this result with the given options, we find that 9 : 24 : 52 is one of the options.

Ratio Concepts Revision Table

Additional Information on Ratios and Proportion

Ratios are closely related to proportions. A proportion is a statement that two ratios are equal.

• Proportion: An equation stating that two ratios are equivalent, e.g., \( \frac{a}{b} = \frac{c}{d} \).
• Direct Proportion: If two quantities are directly proportional, their ratio is constant. As one quantity increases, the other increases at the same rate.
• Inverse Proportion: If two quantities are inversely proportional, their product is constant. As one quantity increases, the other decreases.

Understanding how to manipulate and combine ratios is fundamental in solving many problems involving quantities and their relationships, such as those involving mixtures, scaling, and distribution.