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16 Aug 2021, 08:00
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D
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Difficulty:
65%
(hard)
Question Stats:
64% (02:55) correct
36%
(02:47)
wrong
based on 378
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On a market, two merchants sold total of 100 pineapples and earned the same amount of money. If the first merchant had sold the second merchants pineapples at first merchant's price per pineapple, he would earned $15 and if the second merchant had sold the first merchants pineapples at second merchant's price per pineapple, he would earned \(\$6\frac{2}{3}\). How many pineapples did the second merchant sell ?
A. 20
B. 30
C. 40
D. 60
E. 80
M37-60
A. 20
B. 30
C. 40
D. 60
E. 80
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M37-60
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29 Nov 2021, 05:48
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Official Solution:
On a market, two merchants sold total of 100 pineapples and earned the same amount of money. If the first merchant had sold the second merchants pineapples at first merchant's price per pineapple, he would earned \($15\) and if the second merchant had sold the first merchants pineapples at second merchant's price per pineapple, he would earned \($6\frac{2}{3}\). How many pineapples did the second merchant sell ?
A. \(20\)
B. \(30\)
C. \(40\)
D. \(60\)
E. \(80\)
Say the first merchant sold \(x\) pineapples at \($a\) per pineapple and the second merchant sold \(y\) pineapples at \($b\) per pineapple.
They earned the same amount, so: \(xa = yb\). This gives: \(\frac{a}{b} = \frac{y}{x}\).
If the first merchant had sold the second merchants pineapples at first merchant's price per pineapple, he would earned \($15\): \(ya=15\);
If the second merchant had sold the first merchants pineapples at second merchant's price per pineapple, he would earned \($6\frac{2}{3}\): \(xb=\frac{20}{3}\)
Divide: \(ya=15\) by \(xb=\frac{20}{3}\) to get: \(\frac{y}{x}*\frac{a}{b}=\frac{9}{4}\)
Substitute \(\frac{a}{b} = \frac{y}{x}\) into \(\frac{y}{x}*\frac{a}{b}=\frac{9}{4}\) to get \(\frac{y}{x}*\frac{y}{x}=\frac{9}{4}\);
\((\frac{y}{x})^2=\frac{9}{4}\);
\(\frac{y}{x}=\frac{3}{2}\);
Since \(x+y=100\), then \(x=40\) and \(y=60\).
Answer: D
On a market, two merchants sold total of 100 pineapples and earned the same amount of money. If the first merchant had sold the second merchants pineapples at first merchant's price per pineapple, he would earned \($15\) and if the second merchant had sold the first merchants pineapples at second merchant's price per pineapple, he would earned \($6\frac{2}{3}\). How many pineapples did the second merchant sell ?
A. \(20\)
B. \(30\)
C. \(40\)
D. \(60\)
E. \(80\)
Say the first merchant sold \(x\) pineapples at \($a\) per pineapple and the second merchant sold \(y\) pineapples at \($b\) per pineapple.
They earned the same amount, so: \(xa = yb\). This gives: \(\frac{a}{b} = \frac{y}{x}\).
If the first merchant had sold the second merchants pineapples at first merchant's price per pineapple, he would earned \($15\): \(ya=15\);
If the second merchant had sold the first merchants pineapples at second merchant's price per pineapple, he would earned \($6\frac{2}{3}\): \(xb=\frac{20}{3}\)
Divide: \(ya=15\) by \(xb=\frac{20}{3}\) to get: \(\frac{y}{x}*\frac{a}{b}=\frac{9}{4}\)
Substitute \(\frac{a}{b} = \frac{y}{x}\) into \(\frac{y}{x}*\frac{a}{b}=\frac{9}{4}\) to get \(\frac{y}{x}*\frac{y}{x}=\frac{9}{4}\);
\((\frac{y}{x})^2=\frac{9}{4}\);
\(\frac{y}{x}=\frac{3}{2}\);
Since \(x+y=100\), then \(x=40\) and \(y=60\).
Answer: D
General Discussion
16 Aug 2021, 08:12
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Given: On a market, two merchants sold total of 100 pineapples and earned the same amount of money. If the first merchant had sold the second merchants pineapples at first merchant's price per pineapple, he would earned $15 and if the second merchant had sold the first merchants pineapples at second merchant's price per pineapple, he would earned \(\$6\frac{2}{3}\).
Asked: How many pineapples did the second merchant sell ?
Let the pineapples sold by second merchant be x.
Pineapples sold by first merchant = 100 - x
Let the first merchant's price per pineapple be p1 and second merchant's price per pineapple be p2.
On a market, two merchants sold total of 100 pineapples and earned the same amount of money.
p1 (100-x) = p2 x
p1 /p2 = x/(100-x) (1)
If the first merchant had sold the second merchants pineapples at first merchant's price per pineapple, he would earned $15.
x p1 = 15 (2)
if the second merchant had sold the first merchants pineapples at second merchant's price per pineapple, he would earned \(\$6\frac{2}{3}\)
(100-x)p2 = 6 2/3 = 20/3 (3)
(2)/(3)
xp1/(100-x)p2 = 15*3/20 = 9/4
{x/(100-x)}^2 = 9/4
x/(100 - x) = 3/2
2x = 300 - 3x
5x = 300
x = 300/5 = 60
IMO D
Asked: How many pineapples did the second merchant sell ?
Let the pineapples sold by second merchant be x.
Pineapples sold by first merchant = 100 - x
Let the first merchant's price per pineapple be p1 and second merchant's price per pineapple be p2.
On a market, two merchants sold total of 100 pineapples and earned the same amount of money.
p1 (100-x) = p2 x
p1 /p2 = x/(100-x) (1)
If the first merchant had sold the second merchants pineapples at first merchant's price per pineapple, he would earned $15.
x p1 = 15 (2)
if the second merchant had sold the first merchants pineapples at second merchant's price per pineapple, he would earned \(\$6\frac{2}{3}\)
(100-x)p2 = 6 2/3 = 20/3 (3)
(2)/(3)
xp1/(100-x)p2 = 15*3/20 = 9/4
{x/(100-x)}^2 = 9/4
x/(100 - x) = 3/2
2x = 300 - 3x
5x = 300
x = 300/5 = 60
IMO D
