Problem Solving

BTGmoderatorDC wrote:A credit card company has a cash back option for their customers which is based on the annual amount that they charge on their credit cards. The customer receives 1.5% cash on their first $5000 of purchase, 2.5% on the next $5000 of purchase and a 3% on cash back on portion that exceeds $10000. If a customer receives a cash back of 2.6% on their total annual charges made on the credit card, then what was the customer's annual charge?

1. 22,500

2. 23,500

3. 25,000

4. 26,000

5. 26,500

(1.5% of 5000) + (2.5% of 5000) = 75 + 125 = 200.
We can PLUG IN THE ANSWERS, which represent the total annual charge.
When the correct answer is plugged in, the cash back will be 2.6% of the total annual charge.

D: 26,000
Since 3% is earned on the amount over $10,000 -- in other words, since 3% is earned on $16,000 -- the additional cash back = 3% of 16,000 = 480.
In this case:
(total cash back)/(total annual charge) * 100 = (200+480)/26,000 * 100 = 68/26 = 34/13 ≈ 2.61.
The resulting percentage is just a bit too high.
For the percentage to decrease just a little, the amount that earns 3% must be just a bit lower.
Thus, the total annual charge must be just a bit less than $26,000.

The correct answer is C.

C: 25,000
Since 3% is earned on the amount over $10,000 -- in other words, since 3% is earned on $15,000 -- the additional cash back = 3% of 15,000 = 450.
In this case:
(total cash back)/(total annual charge) * 100 = (200+450)/25,000 * 100 = 65/25 = 13/5 = 2.6.
Success!

Let T be the total spent.

Customer earns $200 on first $10k. So, the total cash back equals:

200 + 0.03(T-10000) = 0.026T
200 + 0.03T - 300 = 0.026T
0.03T = 0.026T + 300
0.004T = 100
T = 25000.

Hence, the correct answer is C. Regards!