If P:Q= 8:15, Q : R = 5: 8 andR : S =4: 5, thenP : Sis equal to

Options

A. `4 :15`
B. `2 :15`
C. `3 :19`
D. `7 :15`

Answer

To solve the problem of finding the ratio \( P:S \) given the ratios \( P:Q = 8:15 \), \( Q:R = 5:8 \), and \( R:S = 4:5 \), we will follow these steps:

### Step 1: Write down the given ratios
We have:
- \( P:Q = 8:15 \)
- \( Q:R = 5:8 \)
- \( R:S = 4:5 \)

### Step 2: Express each variable in terms of a common variable
Let’s express \( P \), \( Q \), \( R \), and \( S \) in terms of a common variable.

1. From \( P:Q = 8:15 \), we can write:
\[ P = 8x \quad \text{and} \quad Q = 15x \]

2. From \( Q:R = 5:8 \), we can express \( R \) in terms of \( Q \):
\[ Q = 5y \quad \text{and} \quad R = 8y \]
Since \( Q = 15x \) and \( Q = 5y \), we can set them equal:
\[ 15x = 5y \implies y = 3x \]
Therefore, substituting \( y \):
\[ R = 8y = 8(3x) = 24x \]

3. From \( R:S = 4:5 \), we can express \( S \) in terms of \( R \):
\[ R = 4z \quad \text{and} \quad S = 5z \]
Since \( R = 24x \) and \( R = 4z \), we can set them equal:
\[ 24x = 4z \implies z = 6x \]
Therefore, substituting \( z \):
\[ S = 5z = 5(6x) = 30x \]

### Step 3: Now we have all variables in terms of \( x \)
- \( P = 8x \)
- \( Q = 15x \)
- \( R = 24x \)
- \( S = 30x \)

### Step 4: Find the ratio \( P:S \)
Now we can find the ratio \( P:S \):
\[ P:S = \frac{P}{S} = \frac{8x}{30x} = \frac{8}{30} \]
We can simplify this:
\[ \frac{8}{30} = \frac{4}{15} \]

### Final Answer
Thus, the ratio \( P:S \) is \( 4:15 \).

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