Calculating Ratios from Given Ratio

The question provides the ratio of two variables, x and y, as x : y = 5 : 2. We are asked to find the value of another ratio involving x and y, specifically (8x + 9y) : (8x + 2y).

Understanding the Given Ratio x : y = 5 : 2

When a ratio x : y is given as 5 : 2, it means that x and y are in proportion to 5 and 2, respectively. We can represent x and y in terms of a common non-zero constant, let's say 'k'.

• x can be written as $x = 5k$
• y can be written as $y = 2k$

Here, k is any non-zero real number.

Substituting Values into the Expression

Now, we need to find the ratio of the expression (8x + 9y) to (8x + 2y). We substitute the values $x = 5k$ and $y = 2k$ into this expression:

The expression is $\frac{8x + 9y}{8x + 2y}$.

Substitute x and y:

Numerator: $8x + 9y = 8(5k) + 9(2k)$

Denominator: $8x + 2y = 8(5k) + 2(2k)$

Simplifying the Expression

Perform the multiplications:

• $8(5k) = 40k$
• $9(2k) = 18k$
• $2(2k) = 4k$

Substitute these results back into the numerator and denominator:

Numerator: $40k + 18k = 58k$

Denominator: $40k + 4k = 44k$

So the ratio becomes $\frac{58k}{44k}$.

Final Ratio Calculation

Since k is a non-zero constant, we can cancel out 'k' from both the numerator and the denominator:

$\frac{58\cancel{k}}{44\cancel{k}} = \frac{58}{44}$

The ratio is currently 58 : 44. To simplify this ratio, we find the greatest common divisor (GCD) of 58 and 44. The GCD of 58 and 44 is 2.

Divide both parts of the ratio by 2:

• $58 \div 2 = 29$
• $44 \div 2 = 22$

The simplified ratio is 29 : 22.

Therefore, the value of (8x + 9y) : (8x + 2y) is 29 : 22.