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2025 AMC 10A Problems/Problem 2: Difference between revisions

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<imath>\textbf{(A)}~3.5\qquad\textbf{(B)}~4\qquad\textbf{(C)}~4.5\qquad\textbf{(D)}~5\qquad\textbf{(E)}~6</imath>
<imath>\textbf{(A)}~3.5\qquad\textbf{(B)}~4\qquad\textbf{(C)}~4.5\qquad\textbf{(D)}~5\qquad\textbf{(E)}~6</imath>


==Solution ==
==Solution 1==
Since the first box had 5 pounds, and 50 percent of it had peanuts, we know there were 5 pounds of peanuts at the beginning. 


Adding the second mixture of nuts, we call this value <imath>x</imath>, as in <imath>x</imath> pounds.
Let the number of pounds of nuts in the second nut mix be <imath>x</imath>. Therefore, we get the equation <imath>0.5 * 10 + 0.2 * x = 0.4(x+10)</imath>. Solving it, we get <imath>x=5</imath>. Therefore the amount of cashews in the two bags is <imath>0.2 * 10 + 0.4 * 5 = 4</imath>, so out answer choice is \boxed{\textbf{(B)} 4}.$
Of that, 20%, or <imath>\frac{x}{5}</imath>, are peanuts.


Since the final percentage is 40 percent peanuts, we have 
~iiiiiizh
<cmath>
\frac{5 + \frac{x}{5}}{10 + x} = \frac{2}{5}.
</cmath>
 
Multiplying both sides by <imath>5(10 + x)</imath>, we get 
<cmath>
25 + x = 20 + 2x.
</cmath>
 
This gives us <imath>x = 5</imath>. 
 
But the problem is asking us to solve for cashews. 
 
The first mixture was <imath>\frac{1}{5}</imath> cashews, so there were <imath>2</imath> pounds of cashews in the first mix. 
In the second, there were <imath>\frac{2x}{5}</imath> cashews, or 2 pounds of cashews. 
 
Adding this together gives us a final total of 
<cmath>
2 + 2 = \boxed{4}
</cmath>
pounds of cashews.
 
*~Minor edits to LaTeX by WildSealVM/Vincent M


==See Also==
==See Also==

Revision as of 16:40, 6 November 2025

The following problem is from both the 2025 AMC 10A #2 and 2025 AMC 12A #2, so both problems redirect to this page.

Problem

A box contains $10$ pounds of a nut mix that is $50$ percent peanuts, $20$ percent cashews, and $30$ percent almonds. A second nut mix containing $20$ percent peanuts, $40$ percent cashews, and $40$ percent almonds is added to the box resulting in a new nut mix that is $40$ percent peanuts. How many pounds of cashews are now in the box?

$\textbf{(A)}~3.5\qquad\textbf{(B)}~4\qquad\textbf{(C)}~4.5\qquad\textbf{(D)}~5\qquad\textbf{(E)}~6$

Solution 1

Let the number of pounds of nuts in the second nut mix be $x$. Therefore, we get the equation $0.5 * 10 + 0.2 * x = 0.4(x+10)$. Solving it, we get $x=5$. Therefore the amount of cashews in the two bags is $0.2 * 10 + 0.4 * 5 = 4$, so out answer choice is \boxed{\textbf{(B)} 4}.$

~iiiiiizh

See Also

2025 AMC 10A (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 10 Problems and Solutions
2025 AMC 12A (Problems • Answer Key • Resources)
Preceded by
Problem 1 		Followed by
Problem 3
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25
All AMC 12 Problems and Solutions

These problems are copyrighted © by the Mathematical Association of America.

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