Question

1. 2 agencies are donating to a communitys fundraising goal. agency 1 donated 1/2 of the fundraising goal, and agency 2 donated 4/5 of the fundraising goal. if the fundraising goal were $7,000 lower, then combined, the agencies would have donated double the fundraising goal. what was the original fundraising goal?

Explanation:

Step1: Define the variable

Let the original fundraising goal be \( x \) dollars.

Step2: Express the donations with the lower goal

If the fundraising goal were \( \$7000 \) lower, the new goal is \( x - 7000 \) dollars.
Agency 1 would donate \( \frac{1}{2}x \) (since the donation is a fraction of the original goal, not the lower one, as the problem says "Agency 1 donated \( \frac{1}{2} \) of the fundraising goal" originally, and we assume the donation fractions are based on the original goal? Wait, no, wait. Wait, the problem says "If the fundraising goal were \( \$7000 \) lower, then combined, the agencies would have donated double the fundraising goal (the lower one)". Wait, let's re - read: "Agency 1 donated \( \frac{1}{2} \) of the fundraising goal, and Agency 2 donated \( \frac{4}{5} \) of the fundraising goal. If the fundraising goal were \( \$7000 \) lower, then combined, the agencies would have donated double the fundraising goal (the new, lower one)". Wait, actually, the donations of the agencies: do they depend on the original goal or the new goal? The problem says "Agency 1 donated \( \frac{1}{2} \) of the fundraising goal" (original), and "Agency 2 donated \( \frac{4}{5} \) of the fundraising goal" (original). So their donations are \( \frac{1}{2}x \) and \( \frac{4}{5}x \) respectively, regardless of the new goal. And the new goal is \( x - 7000 \), and the combined donations equal \( 2(x - 7000) \).

So the equation is: \( \frac{1}{2}x+\frac{4}{5}x=2(x - 7000) \)

Step3: Simplify the left - hand side of the equation

First, find a common denominator for \( \frac{1}{2}x \) and \( \frac{4}{5}x \). The common denominator of 2 and 5 is 10.
\( \frac{1}{2}x=\frac{5}{10}x \) and \( \frac{4}{5}x=\frac{8}{10}x \)
So \( \frac{5}{10}x+\frac{8}{10}x=\frac{5 + 8}{10}x=\frac{13}{10}x \)

Step4: Simplify the right - hand side of the equation

Expand \( 2(x - 7000) \): \( 2x-14000 \)

Step5: Solve the equation

Now we have the equation \( \frac{13}{10}x=2x - 14000 \)
Subtract \( \frac{13}{10}x \) from both sides: \( 0 = 2x-\frac{13}{10}x-14000 \)
Simplify \( 2x-\frac{13}{10}x \): \( 2x=\frac{20}{10}x \), so \( \frac{20}{10}x-\frac{13}{10}x=\frac{7}{10}x \)
The equation becomes \( \frac{7}{10}x=14000 \)
Multiply both sides by \( \frac{10}{7} \): \( x = 14000\times\frac{10}{7} \)
\( 14000\div7 = 2000 \), so \( x=2000\times10 = 20000 \)

Answer:

The original fundraising goal was \(\$20000\)