The ratio of sand to gravel in a mixture is `17 :9` while that between gravel and cement is `6 : 17` . What is the ratio of sand to cement in the mixture ?

Options

A. `17 : 17`
B. `289 : 54`
C. `9 : 6`
D. `2 : 3`

Answer

To find the ratio of sand to cement in the mixture, we can follow these steps:

### Step 1: Understand the given ratios
We have two ratios:
1. The ratio of sand (S) to gravel (G) is \( S : G = 17 : 9 \).
2. The ratio of gravel (G) to cement (C) is \( G : C = 6 : 17 \).

### Step 2: Express the ratios in terms of a common variable
Let's express the quantities in terms of a common variable:
- From the first ratio, we can write:
\[ S = 17x \quad \text{and} \quad G = 9x \]
for some variable \( x \).

- From the second ratio, we can express gravel in terms of cement:
\[ G = 6y \quad \text{and} \quad C = 17y \]
for some variable \( y \).

### Step 3: Set the gravel quantities equal
Since both expressions represent the quantity of gravel (G), we can set them equal to each other:
\[ 9x = 6y \]

### Step 4: Solve for one variable in terms of the other
To find a relationship between \( x \) and \( y \), we can rearrange the equation:
\[ y = \frac{9x}{6} = \frac{3x}{2} \]

### Step 5: Substitute back to find sand and cement
Now, substitute \( y \) back into the expressions for cement (C):
\[ C = 17y = 17 \left(\frac{3x}{2}\right) = \frac{51x}{2} \]

### Step 6: Find the ratio of sand to cement
Now we have:
- \( S = 17x \)
- \( C = \frac{51x}{2} \)

The ratio of sand to cement (S:C) is:
\[ S : C = 17x : \frac{51x}{2} \]

### Step 7: Simplify the ratio
To simplify this ratio, we can multiply both sides by 2 to eliminate the fraction:
\[ S : C = 34x : 51x \]

Now, we can cancel \( x \) from both sides:
\[ S : C = 34 : 51 \]

Next, we can simplify this ratio:
\[ 34 : 51 = \frac{34 \div 17}{51 \div 17} = 2 : 3 \]

### Final Answer
Thus, the ratio of sand to cement in the mixture is \( 2 : 3 \).

---