Understanding the Ratio Problem

The question asks us to find a specific ratio involving three quantities, X, Y, and Z, given two initial ratios relating pairs of these quantities. We are given:

• The ratio of X to Y is 2 : 3, written as \(X : Y = 2 : 3\).
• The ratio of Y to Z is 6 : 7, written as \(Y : Z = 6 : 7\).

We need to calculate the ratio of \((X + Z)\) to Y, which is \((X + Z) : Y\).

Combining Ratios to Find X:Y:Z

To find the ratio \((X + Z) : Y\), we first need to find a combined ratio for X, Y, and Z. This is done by making the common term in the two given ratios equal. The common term is Y.

• In the ratio \(X : Y = 2 : 3\), the value corresponding to Y is 3.
• In the ratio \(Y : Z = 6 : 7\), the value corresponding to Y is 6.

To combine these ratios, we need to make the Y part the same in both. The least common multiple (LCM) of 3 and 6 is 6.

• We need to adjust the ratio \(X : Y = 2 : 3\) so that the Y part becomes 6. To do this, we multiply both parts of the ratio by 2 (since 3 × 2 = 6). \[(2 \times 2) : (3 \times 2) = 4 : 6\] So, \(X : Y\) is equivalent to \(4 : 6\).
• The ratio \(Y : Z = 6 : 7\) already has the Y part as 6.

Now that the value for Y is the same in both adjusted ratios (\(X : Y = 4 : 6\) and \(Y : Z = 6 : 7\)), we can combine them to get the single ratio \(X : Y : Z\).

Thus, \(X : Y : Z = 4 : 6 : 7\).

Calculating the Required Ratio (X + Z) : Y

From the combined ratio \(X : Y : Z = 4 : 6 : 7\), we can represent X, Y, and Z in terms of a common constant, say \(k\), where \(k\) is a non-zero number:

• \(X = 4k\)
• \(Y = 6k\)
• \(Z = 7k\)

Now we can find the value of \((X + Z)\) and then the ratio \((X + Z) : Y\).

• Calculate \(X + Z\): \[X + Z = 4k + 7k = 11k\]
• Now find the ratio \((X + Z) : Y\): \[(X + Z) : Y = 11k : 6k\]

Since \(k\) is a non-zero constant, we can divide both parts of the ratio by \(k\) to simplify it:

\[\frac{11k}{6k} = \frac{11}{6}\]

So, the ratio \((X + Z) : Y\) is \(11 : 6\).

Summary of Steps

1. Identify the given ratios and the ratio to be found.
2. Combine the ratios by making the common term (Y) equal.
3. Use the combined ratio to express X, Y, and Z in terms of a constant.
4. Calculate \((X + Z)\).
5. Form the ratio \((X + Z) : Y\) and simplify it.

Checking the Options

The calculated ratio is \(11 : 6\). Let's compare this with the given options:

• 17 : 14
• 17 : 15
• 11 : 6
• 11 : 8

Our calculated ratio \(11 : 6\) matches one of the options.

Revision Table: Key Ratio Concepts

Additional Information on Ratios

Ratios are often used to express proportional relationships. When we say \(X : Y = 2 : 3\), it means that for every 2 units of X, there are 3 units of Y. This can also be written as a fraction \(\frac{X}{Y} = \frac{2}{3}\). Similarly, \(Y : Z = 6 : 7\) can be written as \(\frac{Y}{Z} = \frac{6}{7}\).

To combine ratios like \(X:Y\) and \(Y:Z\), the key step is to ensure consistency for the variable that appears in both ratios (Y in this case). By finding a common multiple for the Y parts of the ratios, we create a consistent scale that allows us to see the relationship between X, Y, and Z simultaneously.

Ratios can be simplified by dividing all terms by their greatest common divisor (GCD). For example, \(10 : 15\) can be simplified to \(2 : 3\) by dividing both parts by 5.

Understanding how to combine and manipulate ratios is fundamental for solving problems involving proportions, mixtures, sharing quantities, and many other applications in mathematics and real life.