Thread: Find Number of Original Watermelons

1. Find Number of Original Watermelons

The manager at Cream of the Crop bought a load of watermelons for $180. She priced the melons so that she would make$1.50 profit on each melon. When all but 20 had been sold, the manager had recovered her initial investment. How many did she buy originally?

2. Hello, fdrhs1984!

I'll baby-step through the reasoning . . .

The manager at Cream of the Crop bought a load of watermelons for $180. She priced the melons so that she would make$1.50 profit on each melon.
When all but 20 had been sold, the manager had recovered her initial investment.
How many did she buy originally?

She bought $n$ watermelons for $180. The watermelons cost: . $\frac{180}{n}$ dollars each. For a$1.50 profit, she sold them at: . $\frac{180}{n} + 1.5$ dollars each.

She sold $(n-20)$ watermelons and took in $180. There is the equation! . . . . $(n-20)\left(\frac{180}{n} + \frac{3}{2}\right) \:=\: \rlap{///}800 \;\;$ .180 . . . . That's better! ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Expand: . $180 + \frac{3}{2}n - \frac{3600}{n} - 30 \:=\:180 \quad\Rightarrow\quad \frac{3}{2}n - 30 -\frac{3600}{n} \:=\:0$ Multiply by $\frac{2}{3}n\!:\;\;n^2 - 20n - 2400 \:=\:0 \quad\Rightarrow\quad (n-60)(n+40) \:=\:0$ Therefore: . $n \,=\, 60\quad\hdots$ She bought 60 watermelons. ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ ~ Check She bought 60 watermelon for$180.
. . They cost: . $\frac{\180}{60} \:=\:\3$ each.

She charged: . $\3.00 + 1.50 \:=\:\4.50$ for each watermelon.

Then she sold $60-20 \,=\,40$ watermelons
. . and took in: . $40 \times \4.50 \:=\:\180$ (her initial investment)

Nailed it!

3. Soroban really didn't mean to put 800 in his original equation. He was just typing too fast.

There is the equation! . . . . $(n-20)\left(\frac{180}{n} + \frac{3}{2}\right) \:=\:800$

Should've been 180.

Now he must go stand in the corner.

4. I like the way you systematically broke down the word problem to find the answer. This has been my lifelong battle in terms of mathematics.

5. I know Soroban makes very little mistakes (usually typos when he goes too fast).