coin problem

Jan 2008
173
1
I lost my coins! This morning I had 7 coins worth 53 cents. How many nickels (5 cent pieces) did I have?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
 

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MHF Hall of Honor
Mar 2010
2,340
821
Chicago
I lost my coins! This morning I had 7 coins worth 53 cents. How many nickels (5 cent pieces) did I have?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
So I'm assuming we have pennies, nickels, dimes, quarters.

Immediately we can see that there are three pennies. So now you have 4 coins worth 50 cents. Obviously there are no more pennies. Four dimes is not enough to make 50 cents so there must be at least one quarter. And there can't be two quarters because that would already be 50 cents.

So we have 3 coins worth 25 cents. Quick inspection reveals two dimes and one nickel.

So the answer is (A).
 
Dec 2007
3,184
558
Ottawa, Canada
I lost my coins! This morning I had 7 coins worth 53 cents. How many nickels (5 cent pieces) did I have?
(A) 1 (B) 2 (C) 3 (D) 4 (E) 5
May I humbly suggest, Sir Sri340, that you show some effort.
We may soon get tired of doing YOUR work. Understand?
 

Soroban

MHF Hall of Honor
May 2006
12,028
6,341
Lexington, MA (USA)
Hello, sri340!

I have an algebraic solution . . . sort of.


This morning I had 7 coins worth 53 cents.
How many nickels (5 cent pieces) did I have?

. . \(\displaystyle (A)\;1\qquad (B)\;2 \qquad(C)\;3 \qquad (D)\; 4\qquad (E)\; 5\)
Let: .\(\displaystyle \begin{Bmatrix}Q &=&\text{no. of quarters} \\ D &=& \text{no. of dimes} \\ N &=& \text{no. of nickels} \\ P &=& \text{no. of pennies} \end{Bmatrix}\)


\(\displaystyle \begin{array}{ccccc}\text{Their value is 53 cents:} & 25Q + 10D + 5N + P &=& 53 & [1] \\
\text{There are 7 coins:} & Q + D + N + P &=& 7 & [2] \end{array}\)


We see that \(\displaystyle P = 3\)

. . \(\displaystyle \begin{array}{cccccccccc}25Q + 10D + 5N + 3 &=& 53 && \Rightarrow && 5Q + 2D + N &=& 10 & [3] \\
Q + D + N + 3 &=& 7 && \Rightarrow && Q + D + N &=& 4 & [4]\end{array}\)


Subtract [1] - [2]: .\(\displaystyle 4Q + D \:=\:6 \)

We see that: .\(\displaystyle Q \leq 1\), and we know that: .\(\displaystyle D \leq 4\)

. . If \(\displaystyle Q = 0,\:D = 6\) . . . no

. . Hence: .\(\displaystyle Q = 1,\;D = 2\) . . . and \(\displaystyle N = 1\)


You had one nickel . . . answer (A).

 
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