Solving Combined Ratio Problems: Finding N:L

This problem involves combining two given ratios, L:M and M:N, to find a new ratio, N:L. When ratios share a common term, like 'M' in this case, we can combine them by making the value of the common term equal in both ratios.

We are given two ratios:

• Ratio 1: L : M = 3 : 5
• Ratio 2: M : N = 2 : 3

Our goal is to find the ratio N : L.

To combine these ratios, we need to make the value associated with 'M' the same in both ratios. The values for M are 5 and 2. The least common multiple (LCM) of 5 and 2 is 10.

We will adjust each ratio so that the value corresponding to M becomes 10.

For the first ratio, L : M = 3 : 5, multiply both parts by 2 to make the M value 10:

\(L : M = (3 \times 2) : (5 \times 2) = 6 : 10\)

For the second ratio, M : N = 2 : 3, multiply both parts by 5 to make the M value 10:

\(M : N = (2 \times 5) : (3 \times 5) = 10 : 15\)

Now that the value for M is the same in both adjusted ratios (which is 10), we can combine them to find the combined ratio L : M : N:

L : M : N = 6 : 10 : 15

From this combined ratio, we can find the ratio of L to N, which is L : N = 6 : 15.

We can simplify the ratio L : N = 6 : 15 by dividing both terms by their greatest common divisor (GCD), which is 3.

\(L : N = (6 \div 3) : (15 \div 3) = 2 : 5\)

The question asks for the ratio N : L, which is the inverse of L : N.

If L : N = 2 : 5, then N : L = 5 : 2.

Let's summarize the steps:

1. Identify the given ratios and the common term (M).
2. Find the LCM of the values of the common term in the given ratios.
3. Adjust each ratio by multiplying the terms by a factor that makes the common term equal to the LCM.
4. Combine the adjusted ratios to get the combined ratio (L:M:N).
5. Extract the required ratio (N:L) from the combined ratio.
6. Simplify the final ratio if possible.

Comparing our result N : L = 5 : 2 with the given options:

• A. 2 : 1
• B. 5 : 2
• C. 3 : 2
• D. 1 : 2

The calculated ratio N : L = 5 : 2 matches option B.

Revision Table: Key Concepts for Ratio Problems

Additional Information on Ratio and Proportion

Ratios are fundamental in mathematics and are used to show the relative sizes of two or more values. A proportion is an equation stating that two ratios are equal, for example, a:b = c:d, which can also be written as \(\frac{a}{b} = \frac{c}{d}\). Ratio and proportion concepts are widely used in various fields, including scaling recipes, calculating speeds, mixing substances, and understanding geometric similarity.

When dealing with combined ratios like L:M and M:N, the key is ensuring the 'link' (the common term 'M') is consistent across all parts. This method allows us to find the relationship between any two terms in the combined ratio, even if they weren't directly compared in the original given ratios (like finding L:N or N:L from L:M and M:N).

Remember that the order of terms in a ratio matters. L:N is different from N:L. If L:N is 2:5, it means for every 2 units of L, there are 5 units of N. Conversely, N:L = 5:2 means for every 5 units of N, there are 2 units of L.