College

Miriam has 40 crayons and Susie has 72 crayons. If [tex]$x \%$[/tex] of the number of crayons Miriam has is equal to [tex]$y \%$[/tex] of the number of crayons Susie has, what is the ratio of [tex]$x$[/tex] to [tex]$y$[/tex]?

(Note: [tex]$x > 0$[/tex])

Answer :

Sure! Let's solve the given problem step-by-step:

1. Read the Problem:
- Miriam has 40 crayons.
- Susie has 72 crayons.
- We need to find the ratio of [tex]\( x \)[/tex] to [tex]\( y \)[/tex] if [tex]\( x \% \)[/tex] of the number of crayons Miriam has is equal to [tex]\( y \% \)[/tex] of the number of crayons Susie has.

2. Set Up the Equation:
- We are told that [tex]\( x \% \)[/tex] of Miriam's crayons (40) is equal to [tex]\( y \% \)[/tex] of Susie's crayons (72).
- In equation form, it would be:
[tex]\[
\frac{x}{100} \cdot 40 = \frac{y}{100} \cdot 72
\][/tex]

3. Simplify the Equation:
- To simplify, first remove the percentage by multiplying both sides by 100:
[tex]\[
x \cdot 40 = y \cdot 72
\][/tex]
- Rearrange the equation to express one variable in terms of the other:
[tex]\[
x \cdot 40 = y \cdot 72
\][/tex]
- Divide both sides by a common term (in this case, 40 * y):
[tex]\[
\frac{x}{y} \cdot 40 = 72
\][/tex]
- Further simplify:
[tex]\[
\frac{x}{y} = \frac{72}{40}
\][/tex]

4. Simplify the Ratio:
- The fraction [tex]\( \frac{72}{40} \)[/tex] can be simplified by finding the Greatest Common Divisor (GCD) of 72 and 40.
- The GCD of 72 and 40 is 8.
- Divide both the numerator and the denominator by 8:
[tex]\[
\frac{72 \div 8}{40 \div 8} = \frac{9}{5}
\][/tex]

5. Write the Final Ratio:
- Therefore, the ratio [tex]\( \frac{x}{y} \)[/tex] is:
[tex]\[
\frac{9}{5}
\][/tex]

So, the ratio of [tex]\( x \)[/tex] to [tex]\( y \)[/tex] is [tex]\( 9:5 \)[/tex].

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