Understanding Ratio Combination: X : Y and Y : Z

The problem asks us to find the combined ratio of X, Y, and Z, given two separate ratios: X : Y and Y : Z. To combine these ratios, we need to make the common element, which is Y in this case, have the same value in both ratios.

Given Ratios

• Ratio 1: X : Y = $7 : 5$
• Ratio 2: Y : Z = $7 : 11$

Notice that the value for Y is $5$ in the first ratio and $7$ in the second ratio. To combine them into a single X : Y : Z ratio, the value of Y must be consistent.

Method for Combining Ratios

To make the value of Y the same in both ratios, we find the Least Common Multiple (LCM) of the two values of Y, which are $5$ and $7$.

• The LCM of $5$ and $7$ is $5 \times 7 = 35$.

Now, we will adjust each ratio so that the value corresponding to Y becomes $35$.

Adjusting the Ratios

Adjusting Ratio 1 (X : Y = $7 : 5$):

We want the Y value to be $35$. The current Y value is $5$. To get $35$, we multiply $5$ by $7$. We must multiply all parts of the ratio by the same factor to keep the ratio equivalent.

• Multiply X and Y by $7$:
• X : Y = $(7 \times 7) : (5 \times 7)$
• X : Y = $49 : 35$

So, the adjusted first ratio is X : Y = $49 : 35$.

Adjusting Ratio 2 (Y : Z = $7 : 11$):

We want the Y value to be $35$. The current Y value is $7$. To get $35$, we multiply $7$ by $5$. We must multiply all parts of the ratio by the same factor.

• Multiply Y and Z by $5$:
• Y : Z = $(7 \times 5) : (11 \times 5)$
• Y : Z = $35 : 55$

So, the adjusted second ratio is Y : Z = $35 : 55$.

Combining the Adjusted Ratios

Now we have two ratios where the value for Y is the same:

• X : Y = $49 : 35$
• Y : Z = $35 : 55$

Since the Y value is consistently $35$ in both, we can combine them directly to get the ratio X : Y : Z.

X : Y : Z = $49 : 35 : 55$

Final Combined Ratio

The combined ratio of X : Y : Z is $49 : 35 : 55$. We should check if this ratio can be simplified further by finding a common factor for $49$, $35$, and $55$. The factors of $49$ are $1, 7, 49$. The factors of $35$ are $1, 5, 7, 35$. The factors of $55$ are $1, 5, 11, 55$. The only common factor is $1$, so the ratio is already in its simplest form.

Let's quickly verify the original ratios from the combined one:

• X : Y = $49 : 35$. Dividing both by $7$ gives $7 : 5$. This matches the first given ratio.
• Y : Z = $35 : 55$. Dividing both by $5$ gives $7 : 11$. This matches the second given ratio.

The combined ratio is correct.

Therefore, X : Y : Z = $49 : 35 : 55$.

Revision Table: Combining Ratios Concepts

Additional Information on Ratios

Ratios are used to show the relative sizes of two or more values. They are often used in everyday life, such as in cooking recipes, mixing paints, or scaling maps.

• A ratio like $a:b$ can be written as a fraction $\frac{a}{b}$.
• When combining ratios, it's crucial that the common term is placed consistently (e.g., Y is the second term in X:Y and the first term in Y:Z).
• Always simplify ratios to their lowest terms by dividing all parts by their greatest common divisor (GCD).

Understanding how to combine ratios with a common element is a fundamental skill in ratio and proportion problems.