Hello! I am grappling with a mental block rather than a mathematical one, I think, with this question:
"A grocer buys lettuce for $10.40 a crate. Of the lettuce he buys, 10% cannot be sold. If he charges 40˘ for each head he sells and makes a profit of 20˘ on each head he buys, how many heads of lettuce are in the crate?"
"Simple!", I thought. If he charges 0.40 and makes 0.20 in profit, that means that 0.20/head was to buy the heads in the first place, so 10.40/0.20=number of heads. But where does the 10% come in? So I tried to imagine the lettuce coming in, 10% being thrown out, leaving 90% left, or 0.9x heads of lettuce. He makes 0.20 on each one, so profit is 0.2(0.9x). That doesn't help me find out how many heads there were unless that is set equal to something. Equal to what? 10.40? If what he takes in is equal to what he spent, then there's no profit. Okay, then he makes 0.20/head MORE than 10.40, or 10.40+0.2(0.9x). Still not an equal sign in sight.
I know the correct answer (barring a printing error) and through trying different permutations of now-try-this-here I think I know what needs to be multiplied or divided by what to get that correct answer, but I don't understand why it makes sense to do it that way. If you could clarify the how and why about this confoundedly simple-looking problem, I'd greatly appreciate it! Thank you for your time.
EDIT (and correct answer spoiler): After rereading my question a couple times, maybe I understand my lapse. He takes in 0.4(0.9x) total. Half (0.20) is profit, goes into his pocket, and the other 0.20 is offset by costs. Meaning that 0.2(0.9x) DOES equal 10.40, giving 0.18x=10.4, or x=57.7777=58, and 58 is 90% of 65 (10% of 65=6.5, and - rounding up to the nearest whole head of lettuce - 65-7=58), which happens to be the right answer. Am I thinking through this correctly now?
I think an algebraic method, such as the one below should suffice in solving this problem.
Let the number of crates he buys be C, and number of heads in each crate, H
Then he purchases C*H, or CH heads of lettuce
At $10.40 per crate, he purchases C crates, or CH heads of lettuce for $10.4C
Since 10% of a total of CH heads of lettuce could not be sold, then he sells 90% (100% - 10%) of CH heads, or .9CH
Selling .9CH heads at .40c per head means that his proceeds from the sale are: .9CH(.4), or .36CH
Profit equation: Proceeds of sale, less “cost of goods,” equals profit
Proceeds from sale of remaining 90% of entire batch of lettuce: .36CH
Cost of CH heads of lettuce: 10.4C
Profit from sale of each head HE PURCHASES is .20, so profit from TOTAL NUMBER of heads (CH heads)
he PURCHASES = .2(CH), or .2CH
Thus, we get: .36CH – 10.4C = .2CH
.36CH - .2CH = 10.4C
.16CH = 10.4C
8C(.02H) = 8C(1.3) ------- Factoring out GCF, 8C
.02H = 1.3
H, or number of heads of lettuce in each crate = , or
Any questions, or any clarifications, don't hesitate to contact me.
You can do the check!!
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