If P : Q : R = 6 : 7 : 8 and R : S = 3 : 5, then what is P : S?

Options

A. `18:25`
B. `9:20`
C. `11:24`
D. `16:45`

Answer

To find the ratio P : S given the ratios P : Q : R = 6 : 7 : 8 and R : S = 3 : 5, we can follow these steps:

### Step 1: Write down the given ratios
We have:
- P : Q : R = 6 : 7 : 8
- R : S = 3 : 5

### Step 2: Express R in terms of a common value
From the ratio P : Q : R, we can express R as 8x for some value x. Thus:
- P = 6x
- Q = 7x
- R = 8x

### Step 3: Relate R to S using the second ratio
From the ratio R : S = 3 : 5, we can express R as 3y for some value y. Thus:
- R = 3y
- S = 5y

### Step 4: Set the two expressions for R equal to each other
Since both expressions represent R, we can set them equal:
\[ 8x = 3y \]

### Step 5: Solve for y in terms of x
To find a relationship between x and y, we can rearrange the equation:
\[ y = \frac{8x}{3} \]

### Step 6: Substitute y back into the expression for S
Now we can substitute y back into the expression for S:
\[ S = 5y = 5 \left(\frac{8x}{3}\right) = \frac{40x}{3} \]

### Step 7: Now we have P and S in terms of x
We have:
- P = 6x
- S = \frac{40x}{3}

### Step 8: Find the ratio P : S
Now we can find the ratio P : S:
\[ P : S = 6x : \frac{40x}{3} \]

### Step 9: Simplify the ratio
To simplify, we can multiply both sides by 3 to eliminate the fraction:
\[ P : S = 6x \cdot 3 : 40x = 18x : 40x \]

### Step 10: Cancel out x
Since x is common in both terms, we can cancel it out:
\[ P : S = 18 : 40 \]

### Step 11: Simplify the ratio further
Now we can simplify 18 : 40 by dividing both terms by 2:
\[ P : S = 9 : 20 \]

### Final Answer
Thus, the ratio P : S is **9 : 20**.

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