To determine the minimum number of additional successful free throws Sherita needs to make in order to have a free-throw average of at least 80%, we can set up and solve an algebraic equation. Here's the step-by-step process:
1. Define Variables:
- Let [tex]\( x \)[/tex] be the number of additional free throws Sherita needs to make.
2. Current Stats:
- Sherita has made 46 free throws.
- Sherita has attempted 60 free throws.
3. Target Average:
- The target free-throw average is 80%.
4. Set Up the Equation:
- The new number of free throws made will be [tex]\( 46 + x \)[/tex].
- The new number of free throws attempted will be [tex]\( 60 + x \)[/tex].
- To have an average of at least 80%, the ratio of made free throws to attempted free throws should be at least 0.80:
[tex]\[
\frac{46 + x}{60 + x} \geq 0.80
\][/tex]
5. Solve the Equation:
- Start by isolating [tex]\( x \)[/tex] through algebraic manipulation:
[tex]\[
\frac{46 + x}{60 + x} \geq 0.80
\][/tex]
Multiply both sides by [tex]\( 60 + x \)[/tex] to clear the fraction:
[tex]\[
46 + x \geq 0.80(60 + x)
\][/tex]
Distribute the 0.80 on the right-hand side:
[tex]\[
46 + x \geq 48 + 0.80x
\][/tex]
Subtract [tex]\( 0.80x \)[/tex] from both sides:
[tex]\[
46 + x - 0.80x \geq 48
\][/tex]
Simplify the left-hand side:
[tex]\[
46 + 0.20x \geq 48
\][/tex]
Subtract 46 from both sides:
[tex]\[
0.20x \geq 2
\][/tex]
Divide both sides by 0.20:
[tex]\[
x \geq \frac{2}{0.20}
\][/tex]
[tex]\[
x \geq 10
\][/tex]
6. Conclusion:
- Sherita needs to make at least 10 additional free throws to reach a free-throw average of at least 80%.
Thus, the minimum number of free throws Sherita would need to make from now on to achieve a free-throw average of at least 80% is:
[tex]\[ \boxed{10} \][/tex]
