Sure, let's find out how many cups of sugar are needed to make 14 cookies when you know that [tex]$1 \frac{1}{4}$[/tex] cups of sugar are needed to make 20 cookies.
### Step-by-Step Solution:
1. Convert Mixed Number to Improper Fraction:
- You need [tex]$1 \frac{1}{4}$[/tex] cups of sugar for 20 cookies.
- Convert [tex]$1 \frac{1}{4}$[/tex] to an improper fraction:
[tex]\[
1 \frac{1}{4} = 1 + \frac{1}{4} = \frac{4}{4} + \frac{1}{4} = \frac{5}{4}
\][/tex]
2. Proportionality Setup:
- We need to find out how many cups of sugar are required for 14 cookies.
- Since the ratio of sugar to cookies should remain the same, set up the proportion:
[tex]\[
\frac{\text{sugar for 20 cookies}}{20} = \frac{\text{sugar for 14 cookies}}{14}
\][/tex]
We know the sugar for 20 cookies is [tex]$\frac{5}{4}$[/tex] cups. Let [tex]\( x \)[/tex] be the cups of sugar needed for 14 cookies:
[tex]\[
\frac{\frac{5}{4}}{20} = \frac{x}{14}
\][/tex]
3. Solve for [tex]\( x \)[/tex]:
- To isolate [tex]\( x \)[/tex], first simplify the left fraction:
[tex]\[
\frac{\frac{5}{4}}{20} = \frac{5}{4} \times \frac{1}{20} = \frac{5}{80} = \frac{1}{16}
\][/tex]
- Now, the equation looks like this:
[tex]\[
\frac{1}{16} = \frac{x}{14}
\][/tex]
- Solve for [tex]\( x \)[/tex] by cross-multiplying:
[tex]\[
x = 14 \times \frac{1}{16} = \frac{14}{16}
\][/tex]
- Simplify the fraction:
[tex]\[
\frac{14}{16} = \frac{7}{8}
\][/tex]
So, you will need [tex]\(\frac{7}{8}\)[/tex] cups of sugar to make 14 cookies.
