The ratio of five-dollar bills to one-dollar bills in Armando’s wallet

Given

• The ratio of five-dollar bills to one-dollar bills in Armando’s wallet is 2:3.
• He made a $7 purchase, he paid for that purchase with only five-dollar bills, and with the smallest possible number of five-dollar bills.
• He received his change in one-dollar bills. After completing that transaction.
• The ratio of five-dollar bills to one-dollar bills was 1: 3.

To Find

• The number of one-dollar bills did Armando have before his purchase.

Approach and Working Out

• The number of five dollars = F, the number of one-dollar bills = O

o F : O = 2 : 3
o F = 2x, O = 3x

• After the purchase he must have given 2 Five dollar bills and got 3 one-dollar bills back.

o Five dollar bills now 2x – 2
o One dollar bills now 3x + 3

• \(\frac{(2x – 2)}{(3x + 3)}\) = \(\frac{1}{3}\)

o 6x – 6 = 3x + 3
o 3x = 9
o x = 3
o One dollar bill = 3x = 9

Correct Answer: Option D

2x-2/3x+3=1/3
x=3
3x=3(3)=9

ans (D)

The ratio of five-dollar bills to one-dollar bills in Armando’s wallet is 2:3

Given Information 1:
7$ purchase paid with smallest possible 5 dollar bills and also received the change.

This shows that he paid with 2 5$ dollar bills and received 3 $1 bills as change.

Given Information 2:
After completing that transaction, the ratio of five-dollar bills to one-dollar bills was 1: 3

This shows that Armando had 5$ bills after completing the transaction and from initial ratio, the number of 5$ bills has to be a multiple of 2.

So 2x be the initial number of 5$ bills armando had and he gave away 2, we get 2x-2
Also initially he had 3 1$ bills and gained additional 3 1$ bills after purchase, so we get 3x+3

And the new ratio is 1/3

solving \(\frac{(2x-2)}{ (3x+3)} = \frac{1}{3}\)

cross multiple 3*(2x-2)=3x+3
X = 9

Ans: D

Before purchase :

No of 1$ Bill = 3x
No of 5$ Bill = 2x

After purchase :

2 5$ bill are given and 3 1$ bill are recieved

No of 1$ Bill = 3x + 3
No of 5$ Bill = 2x - 2

Given 2x - 2 : 3x + 3 = 1 : 3

After solving , X = 3

No of 1$ bill before purchase = 3X = 9

IMO D

ratio : $5 to $1 - 2 : 3 = 2x:3x

Used 2 5$ and got 3 1$ change.

New ratio : ratio : $5 to $1 -1:3


Solving, we get 3*(2x - 2) = 3x + 3

6x - 6 = 3x + 3

3x = 9 or x = 3

Number of $1 bills = 3x = 3 * 3 = 9

Solution:

Since the ratio of five-dollar bills to one-dollar bills in Armando’s wallet is 2:3 (before he made the purchase), the number of one-dollar bills he had must be a multiple of 3. Therefore, of the given answer choices, only 6 and 9 could be the correct answer. Let’s check these two answer choices.

If there were 6 one-dollar bills, there were 4 five-dollar bills. Since he had to pay the purchase of $7 with 2 five-dollars, he would receive 3 one-dollar bills as the change. Therefore, after the purchase, he had 2 five-dollar bills and 9 one-dollar bills,yielding a ratio of 2 : 9. However, since this ratio is not 1:3, he couldn’t have 6 one-dollar bills in his wallet before the purchase. Thus, this leaves 9 as the only possible correct answer.

Answer: D

A simple arithmetic question dealing with Ratios.

Given: The ratio of five-dollar bills to one-dollar bills in Armando’s wallet is 2:3.

\(\frac{X}{Y} = \frac{2}{3}\)

\(X = \frac{2Y}{3}\)

Given: He made a 7$ purchase with only 5$ bills.
So, he would have used 2 5$ bills and would have received 3 1$ bills in return and now the new ratio is 1/3

\(\frac{X-2}{Y+3} = \frac{1}{3}\)

\((X-2)(3) = Y+3\)

\(3X - 6 = Y + 3\)

\(3X = Y + 9\)

Substitute the value of \(X = \frac{2Y}{3}\)

\(3\frac{2Y}{3} = Y + 9\)

\(Y = 9\)

Option (E) is the correct answer


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My 2 cents on this:

Others above have solved the equations to the tea, but I just want everybody to remember the last sentence of the question How many one-dollar bills did Armando have before his purchase?

Don't be in haste to pick A! (which is five dollar bills) Answer is D, 9 one dollar bills.