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06 Aug 2026, 06:08
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A city council distributed a $320,000 grant among exactly three programs: Youth Sports, Public Gardens, and Senior Services. Did any one of the three programs receive more than 37.5% of the total grant?
(1) Senior Services received more funding than each of the other two programs.
(2) Youth Sports received 1.5 times as much funding as Public Gardens.
(1) Senior Services received more funding than each of the other two programs.
(2) Youth Sports received 1.5 times as much funding as Public Gardens.
06 Aug 2026, 06:08
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Official Solution:
A city council distributed a $320,000 grant among exactly three programs: Youth Sports, Public Gardens, and Senior Services. Did any one of the three programs receive more than \(37.5\%\) of the total grant?
\(37.5\%\) of 320,000 is \(0.375 * 320,000 = 120,000\).
So, the question asks whether any program received more than $120,000.
(1) Senior Services received more funding than each of the other two programs.
This tells us only that Senior Services received the largest amount.
For example, Senior Services could have received $300,000, and the answer would be YES.
But Senior Services could have received $115,000, while the other two programs received $105,000 and $100,000, and the answer would be NO.
Not sufficient.
(2) Youth Sports received 1.5 times as much funding as Public Gardens.
This gives only the relationship between two programs.
For example:
• Public Gardens \(= 80,000\)
• Youth Sports \(= 120,000\)
• Senior Services \(= 120,000\)
No program received more than $120,000.
But:
• Public Gardens \(= 60,000\)
• Youth Sports \(= 90,000\)
• Senior Services \(= 170,000\)
One program received more than $120,000.
Not sufficient.
(1)+(2) Let Public Gardens receive \(x\).
Then Youth Sports received \(1.5x\).
Since Senior Services received more funding than each of the other two programs, Senior Services received more than \(1.5x\). So let Senior Services receive \(1.5x + k\), where \(k > 0\).
Then the total grant was:
\(x + 1.5x + 1.5x + k = 4x + k\)
Senior Services’ share of the total grant was:
\(\frac{1.5x + k}{4x + k}\)
We need to know whether this is greater than \(37.5\%\), or \(\frac{3}{8}\).
Is \(\frac{1.5x + k}{4x + k} > \frac{3}{8}\)?
Is \(8(1.5x + k) > 3(4x + k)\)?
Is \(12x + 8k > 12x + 3k\)?
Is \(5k > 0\)?
Since \(k > 0\), the answer is YES.
So Senior Services received more than \(37.5\%\) of the grant.
Sufficient.
Answer: C
A city council distributed a $320,000 grant among exactly three programs: Youth Sports, Public Gardens, and Senior Services. Did any one of the three programs receive more than \(37.5\%\) of the total grant?
\(37.5\%\) of 320,000 is \(0.375 * 320,000 = 120,000\).
So, the question asks whether any program received more than $120,000.
(1) Senior Services received more funding than each of the other two programs.
This tells us only that Senior Services received the largest amount.
For example, Senior Services could have received $300,000, and the answer would be YES.
But Senior Services could have received $115,000, while the other two programs received $105,000 and $100,000, and the answer would be NO.
Not sufficient.
(2) Youth Sports received 1.5 times as much funding as Public Gardens.
This gives only the relationship between two programs.
For example:
• Public Gardens \(= 80,000\)
• Youth Sports \(= 120,000\)
• Senior Services \(= 120,000\)
No program received more than $120,000.
But:
• Public Gardens \(= 60,000\)
• Youth Sports \(= 90,000\)
• Senior Services \(= 170,000\)
One program received more than $120,000.
Not sufficient.
(1)+(2) Let Public Gardens receive \(x\).
Then Youth Sports received \(1.5x\).
Since Senior Services received more funding than each of the other two programs, Senior Services received more than \(1.5x\). So let Senior Services receive \(1.5x + k\), where \(k > 0\).
Then the total grant was:
\(x + 1.5x + 1.5x + k = 4x + k\)
Senior Services’ share of the total grant was:
\(\frac{1.5x + k}{4x + k}\)
We need to know whether this is greater than \(37.5\%\), or \(\frac{3}{8}\).
Is \(\frac{1.5x + k}{4x + k} > \frac{3}{8}\)?
Is \(8(1.5x + k) > 3(4x + k)\)?
Is \(12x + 8k > 12x + 3k\)?
Is \(5k > 0\)?
Since \(k > 0\), the answer is YES.
So Senior Services received more than \(37.5\%\) of the grant.
Sufficient.
Answer: C
