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Zorluk: ZorPercent Change and Interest

An educational foundation set an annual fundraising goal for a given year. During the first quarter, the foundation raised 30%30\% of its annual goal. In the second quarter, the amount raised was 25%25\% greater than the amount raised in the first quarter. In the third quarter, the amount raised was 40%40\% less than the total amount raised in the first two quarters combined. If the total amount raised across the first three quarters combined was $129,600\$129,600, what was the foundation's annual fundraising goal, in dollars?

Cevap: 120000 dollars

Cevap

The foundation's annual fundraising goal was $120,000.
To determine the annual goal GG, calculate the raised amount in each quarter relative to GG. The first quarter yields Q1=0.30GQ_1 = 0.30G. The second quarter amount is 25%25\% greater than Q1Q_1, which equals 1.25×0.30G=0.375G1.25 \times 0.30G = 0.375G. The sum of the first two quarters is 0.30G+0.375G=0.675G0.30G + 0.375G = 0.675G. The third quarter amount is 40%40\% less than this combined sum, giving 0.60×0.675G=0.405G0.60 \times 0.675G = 0.405G. Summing the three quarters gives a total of 0.30G+0.375G+0.405G=1.08G0.30G + 0.375G + 0.405G = 1.08G. Equating 1.08G=129,6001.08G = 129,600 and dividing by 1.081.08 gives G=120,000G = 120,000.

Adım Adım Çözüm

1
Define the variable for the unknown quantity
Let GG represent the foundation's annual fundraising goal in dollars.
Establishing a variable allows all quarterly contributions to be expressed as a linear algebraic function of the goal.
2
Express the amount raised in the first quarter in terms of GG
Q1=0.30GQ_1 = 0.30G
The foundation raised 30%30\% of its annual goal during the first quarter.
3
Express the amount raised in the second quarter in terms of GG
Q2=1.25×0.30G=0.375GQ_2 = 1.25 \times 0.30G = 0.375G
An increase of 25%25\% over Q1Q_1 means Q2=(1+0.25)Q1=1.25×0.30GQ_2 = (1 + 0.25) Q_1 = 1.25 \times 0.30G.
4
Calculate the combined total raised in the first two quarters
Q1+Q2=0.30G+0.375G=0.675GQ_1 + Q_2 = 0.30G + 0.375G = 0.675G
This combined sum serves as the base value for calculating the third quarter's contribution.
5
Express the amount raised in the third quarter in terms of GG
Q3=(1−0.40)×0.675G=0.60×0.675G=0.405GQ_3 = (1 - 0.40) \times 0.675G = 0.60 \times 0.675G = 0.405G
The third quarter raised 40%40\% less than the combined amount of the first two quarters, meaning it equaled 60%60\% of (Q1+Q2)(Q_1 + Q_2).
6
Sum the contributions of all three quarters and solve for GG
Total =0.30G+0.375G+0.405G=1.08G=129,600  ⟹  G=120,000= 0.30G + 0.375G + 0.405G = 1.08G = 129,600 \implies G = 120,000
Setting the sum equal to the total dollar amount raised ($129,600\$129,600) yields 1.08G=129,6001.08G = 129,600, so G=129,6001.08=120,000G = \frac{129,600}{1.08} = 120,000.

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