Question

For three consecutive years, Sam invested some money at the start of the year. The first year, he invested x dollars. The second year, he invested $2,000 less than 5/2 times the amount he invested the first year. The third year, he invested $1,000 more than 1/5 of the amount he invested the first year.

During the same three years, Sally also invested some money at the start of every year. The first year, she invested $1,000 less than 3/2 times the amount Sam invested the first year. The second year, she invested $1,500 less than 2 times the amount Sam invested the first year. The third year, she invested $1,400 more than 1/4 of the amount Sam invested the first year.

If Sam and Sally invested the same total amount at the end of three years, the amount Sam invested the first year is $ ???, and the amount Sally invested the last year is $ ???

Answer 1

The amount Sam invested the first year is $2000 and the amount Sally invested the last year is $1900.

Answer 2

The first year Sam invested $2,000. The third year sally invested $1,900.

Explanation:

Let $x be the amount of money Sam invested the first year. The second year, he invested $2,000 less than 5/2 times the amount he invested the first year, then the second year he invested

`\$\left((5)/(2)x-2,000\right).`

The third year, he invested $1,000 more than 1/5 of the amount he invested the first year, then the third year he invested

`\$\left((1)/(5)x+1,000\right).`

During three years Sam invested

`\$\left(x+(5)/(2)x-2,000+(1)/(5)x+1,000\right)=\$\left((37)/(10)x-1,000\right).`

The first year, Sally invested $1,000 less than 3/2 times the amount Sam invested the first year, then the first year she invested

`\$\left((3)/(2)x-1,000\right).`

The second year, she invested $1,500 less than 2 times the amount Sam invested the first year, then the second year she invested

`\$(2x-1,500).`

The third year, she invested $1,400 more than 1/4 of the amount Sam invested the first year, then the third year she invested

`\$\left((1)/(4)x+1,400\right).`

During three years Sally invested

`\$\left((3)/(2)x-1,000+2x-1,500+(1)/(4)x+1,400\right)=\$\left((15)/(4)x-1,100\right).`

If Sam and Sally invested the same total amount at the end of three years, then

`(37)/(10)x-1,000=(15)/(4)x-1,100,\\ \\(37)/(10)x-(15)/(4)x=1,000-1,100,\\ \\-(1)/(20)x=-100,\\ \\x=\$2,000.`

Thus,

`\$\left((1)/(4)\cdot 2,000+1,400\right)=\$1,900.`