Understanding Ratio and Proportion

A ratio is a comparison of two quantities of the same kind, expressed as a : b or \frac{a}{b}. A proportion is an equality of two ratios.

In this problem, we are given three pairwise ratios: M : N, N : O, and O : P. Our goal is to find the combined proportion M : N : O : P.

Combining Given Ratios Step-by-Step

We are given the following ratios:

• M : N = 8 : 15
• N : O = 5 : 9
• O : P = 18 : 19

To find the combined ratio M : N : O : P, we need to make the common terms in the adjacent ratios equal. We will combine M : N and N : O first, and then combine the resulting M : N : O with O : P.

Step 1: Combine M : N and N : O

We have M : N = 8 : 15 and N : O = 5 : 9. The common term is N. The values of N in these ratios are 15 and 5. To combine these, we need to make the value of N the same in both ratios. The least common multiple (LCM) of 15 and 5 is 15.

The first ratio M : N = 8 : 15 already has N = 15.

For the second ratio N : O = 5 : 9, we need to make N = 15. We can achieve this by multiplying both parts of the ratio by \frac{15}{5} = 3.

N : O = (5 \times 3) : (9 \times 3) = 15 : 27

Now we have M : N = 8 : 15 and N : O = 15 : 27. Since the value of N is the same (15), we can combine them to get the ratio M : N : O.

M : N : O = 8 : 15 : 27

Step 2: Combine M : N : O and O : P

We now have M : N : O = 8 : 15 : 27 and the third ratio O : P = 18 : 19. The common term between M : N : O and O : P is O. The values of O are 27 (from M : N : O) and 18 (from O : P). To combine these, we need to make the value of O the same in both cases. The LCM of 27 and 18 is 54.

For the ratio M : N : O = 8 : 15 : 27, we need to make O = 54. We multiply all parts of this ratio by \frac{54}{27} = 2.

M : N : O = (8 \times 2) : (15 \times 2) : (27 \times 2) = 16 : 30 : 54

For the ratio O : P = 18 : 19, we need to make O = 54. We multiply both parts of this ratio by \frac{54}{18} = 3.

O : P = (18 \times 3) : (19 \times 3) = 54 : 57

Now we have M : N : O = 16 : 30 : 54 and O : P = 54 : 57. Since the value of O is the same (54), we can combine them to get the final combined proportion M : N : O : P.

M : N : O : P = 16 : 30 : 54 : 57

Final Combined Proportion

The combined proportion M : N : O : P is 16 : 30 : 54 : 57. We can verify this:

• M : N = 16 : 30. Dividing by 2, we get 8 : 15. (Matches the given M : N = 8 : 15)
• N : O = 30 : 54. Dividing by 6, we get 5 : 9. (Matches the given N : O = 5 : 9)
• O : P = 54 : 57. Dividing by 3, we get 18 : 19. (Matches the given O : P = 18 : 19)

The derived ratio 16 : 30 : 54 : 57 is consistent with the given ratios.

Comparing with Options

Let's compare our calculated combined ratio 16 : 30 : 54 : 57 with the provided options:

Option 4 matches our calculated combined proportion.

Revision Table: Key Ratio Concepts

Additional Information on Combining Ratios

Combining ratios is a common technique in ratio and proportion problems. It's essential when you have relationships between multiple variables given in pairs, like A:B, B:C, C:D, and you need to find the overall relationship A:B:C:D.

The core idea is to find a common value for the connecting term(s) in the ratios. For example, when combining M:N and N:O, the connecting term is N. We find the LCM of the values of N in both ratios and scale each ratio appropriately so that the N values become equal to the LCM. Once the connecting terms are equal, the ratios can be written together as a single combined ratio.

This method can be extended to any number of ratios that form a chain, such as A:B, B:C, C:D, etc., to find A:B:C:D:....

Alternatively, ratios can be expressed as fractions and multiplied, though the LCM method is often clearer for visualizing the combined proportion directly.