A purse contains 5-cent coins and 10-cent coins worth a total of $1.75

Question

A purse contains 5-cent coins and 10-cent coins worth a total of $1.75. If the 5-cent coins were replaced with 10-cent coins and the 10-cent coins were replaced with 5-cent coins, the coins would be worth a total of $2.15. How many coins are in the purse?

A. 26
B. 27
C. 28
D. 29
E. 30

A. 26
B. 27
C. 28
D. 29
E. 30

vishalbalwani · Accepted Answer

Let the number of  5 cent coins be a  and the number of  10 cent coins be b

we have been give  5a + 10b = 175  (total value in cents)
on reversing the numbers we get  5b + 10a = 215  (total value in cents)
add both the equations to get   15(a + b) = 390 
therefore a + b = 26

Answer is (A)

pushpitkc · Answer

Let the total 5 cent coins be x, and the 10 cent coins be y
 
From the question stem,  5x + 10y = 175 
Value of y(by the following equation) :  y = \frac{175 - 5x}{10}

Substituting this value of y in the second equation
 10x + \frac{5(175 - 5x)}{10} = 215 
 100x + 875 - 25x = 2150 
 75x = 1275 
 x = 17

Substituting this value of x in equation 5x+10y = 175 
 85 + 10y = 175 
 y = 9

Therefore the sum of coins in the p = 17 + 9 =  26(Option A)

generis · Answer

I solved with two variables, and it works, but I have a question about my own method that applies to  vishalbalwani  's method, too.

Let A = the number of 5-cent coins
Let B = the number of 10- cent coins

In one scenario, the coins, in their respective unknown quantities, total $1.75

In a second scenario, the 5-cent coins and 10-cent coins exchange quantities exactly, and they total $2.15

5A + 10B = 175 (P)
10A + 5B = 215 (Q)

Multiply (Q) by two, and subtract (P)

20 A + 10B = 430
 __5A + 10B = 175 
15A = 255
A = 17

There are 17 coins worth 5 cents.

Use (P) to find the number of 10-cent pieces

5(17) + 10B = 175
10B = 90
B = 9
There are 9 coins worth 19
0 cents.

A = 17, B = 9, total = 26 coins

Answer A

I have run this problem three ways, including  vishalbalwani  's, and working from answer choices. This combination (number of coins), is the only one that works.

Question: Is the method sound? The variables work, but they seem inconsistent.

I use A as a quantity for 5-cent coins. The coefficients of the variables -- 5 and 10 -- are the  values  of the coins in cents.

But in the second equation, (Q) I do not think I have switched quantities, per the prompt. I think have switched values. In the second equation, (Q), the 5, a value, is in front of B -- which is supposed to be the  quantity  of 10-cent coins.  vishalbalwani  did the same.

What am I missing? We are supposed to be switching quantities. But switching values works.

I think  pushpitkc  avoids the whole problem by immediately defining y in terms of x from one equation.

Is my method sound?

ScottTargetTestPrep · Answer

We can create two equations in which a = the initial number of 5-cent coins and b = the initial number of 10-cent coins. We can create the “money” equation by recalling that a 5-cent coin is $0.05 and a 10-cent coin is $0.10; thus:

0.05a + 0.1b = 1.75

And, after we reverse the coinage, we have:

0.1a + 0.05b = 2.15

Multiplying the first equation by -2, we have:

-0.1a - 0.2b = -3.50

Adding the two equations, we have:

-0.15b = -1.35

b = 9

Thus:

0.05a + (0.1)(9) = 1.75

0.05a = 0.85

a = 17

So, there is a total of 17 + 9 = 26 coins.

Answer: A

generis · Answer

ScottTargetTestPrep  , where you reverse the coinage, what do variables  a  and  b  stand for? (I had the same question about my own method, above.)

Perhaps the variables stand simply for the total number of coins, and absent any total number of coins, we cannot write one variable in terms of another.

Conceptually, however, that reasoning does not make a lot of sense to me.

We have two unknown quantities in two scenarios. In the first equation, the quantity b is defined as the number of $0.10 coins (and qty "a" initially is # of $.05 coins)

If "b" then switches to the number of $0.05 coins (vice versa for "a"), how are those two a's and b's the same, such that we can use them as an identity in solving two equations?

Sorry if this question is sophomoric; usually I can find my way out of a muddle. Not this time.

chetan2u · Answer

Hi....

A logical way...

When you interchange, the total amount increases, so 5-cents are more.
How much ? Say x, so 5x=215-175=40... X=8...

So initial amount is 5(y+8)+10y=175.....15y+40=175.....15y=135....y=9
Total coins = y+8+y=9+8+9=26

A

ScottTargetTestPrep · Answer

The variables a and b stand for the initial number of 5- and 10-cent coins, before and after reversing the coinage. The only thing that changes when the coinage is reversed is that the number of 5-cent coins is now b and the number of 10-cent coins is now a.

bumpbot · Answer

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A purse contains 5-cent coins and 10-cent coins worth a total of $1.75

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