To solve for the number that makes the ratio equivalent to [tex]\( 7:8 \)[/tex] when the new ratio's denominator is 48, follow these detailed steps:
1. Understand the Given Ratio:
The given ratio is [tex]\( 7:8 \)[/tex]. This means for every 7 parts of something, there are 8 parts of something else.
2. Establish the New Ratio's Denominator:
We are told that the new ratio's denominator is 48. Let's denote the unknown numerator by [tex]\( x \)[/tex].
3. Set Up the Proportion:
Since the two ratios are equivalent, we can set up the proportion as follows:
[tex]\[
\frac{7}{8} = \frac{x}{48}
\][/tex]
4. Solve for [tex]\( x \)[/tex]:
To find the value of [tex]\( x \)[/tex] that makes the ratios equivalent, we solve the proportion. We can do this by cross-multiplying:
[tex]\[
7 \times 48 = 8 \times x
\][/tex]
5. Perform the Multiplications:
Calculate the left hand side of the equation:
[tex]\[
7 \times 48 = 336
\][/tex]
6. Isolate [tex]\( x \)[/tex]:
Now, isolate [tex]\( x \)[/tex] by dividing both sides of the equation by 8:
[tex]\[
x = \frac{336}{8}
\][/tex]
7. Divide and Find [tex]\( x \)[/tex]:
Perform the division:
[tex]\[
x = 42
\][/tex]
Therefore, the number that makes the ratio [tex]\( 7:8 \)[/tex] equivalent to [tex]\( x:48 \)[/tex] is [tex]\( 42 \)[/tex]. Hence, the new ratio is [tex]\( 42:48 \)[/tex].
