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For three consecutive years, Sam and Sally invested money at the start of each year.

- The first year, Sam invested [tex]$x$[/tex].
- The second year, he invested half the amount he invested the first year.
- The third year, he invested [tex]$1,000 more than \(\frac{1}{5}\) of the amount he invested the first year.

- The first year, Sally invested $[/tex]y[tex]$.
- The second year, she invested $[/tex]1,500 less than 2 times the amount Sam invested the first year.
- The third year, she invested the same amount as 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 [tex]$\boxed{\text{ }}$[/tex].



Answer :

GinnyAnswer
Let's solve this step by step.

1. Let [tex]\( x \)[/tex] be the amount Sam invested the first year.

2. Investments made by Sam over the three years:
- First year: [tex]\( x \)[/tex]
- Second year: [tex]\( 2x \)[/tex]
- Third year: [tex]\( \frac{x}{5} + 1000 \)[/tex]

3. Investments made by Sally over the three years:
- First year: [tex]\( 2x \)[/tex]
- Second year: [tex]\( 2x - 1500 \)[/tex]
- Third year: [tex]\( x \)[/tex]

4. Calculate the total investment by Sam over the three years:
- Total investment by Sam [tex]\( = x + 2x + \left(\frac{x}{5} + 1000\right) \)[/tex]

5. Calculate the total investment by Sally over the three years:
- Total investment by Sally [tex]\( = 2x + (2x - 1500) + x \)[/tex]

6. Equate the total investments since Sam and Sally invested the same total amount:
[tex]\[ x + 2x + \left(\frac{x}{5} + 1000\right) = 2x + (2x - 1500) + x \][/tex]

7. Simplify the equation to find [tex]\( x \)[/tex]:
- Left side: [tex]\( x + 2x + \frac{x}{5} + 1000 = \left(\frac{5x}{5} + \frac{10x}{5} + \frac{x}{5}\right) + 1000 = \frac{16x}{5} + 1000 \)[/tex]
- Right side: [tex]\( 2x + 2x - 1500 + x = 5x - 1500 \)[/tex]

- Equate the two sides:
[tex]\[ \frac{16x}{5} + 1000 = 5x - 1500 \][/tex]

- Multiply everything by 5 to clear the fraction:
[tex]\[ 16x + 5000 = 25x - 7500 \][/tex]

- Rearrange the terms to solve for [tex]\( x \)[/tex]:
[tex]\[ 5000 + 7500 = 25x - 16x \][/tex]
[tex]\[ 12500 = 9x \][/tex]
[tex]\[ x = \frac{12500}{9} \][/tex]

Hence, the amount Sam invested the first year is [tex]\( \boxed{\frac{12500}{9}} \)[/tex] dollars.

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