Question Details

An electronics distributor placed an order for 10 crates of premium monitors alongside several crates of standard keyboards. Each crate of premium monitors costs double the price of a crate of standard keyboards. Because of a shipping processing error, the numbers of monitor crates and keyboard crates were interchanged, which resulted in a 20% increase in the total order cost. What was the initial ratio of monitor crates to keyboard crates ordered?

Options

3:2

1:3

4:7

2:3

Show Answer

Correct Answer :

Option C

4:7

Solution :

The correct answer is 4:7.

Let us solve this problem step-by-step by setting up cost equations based on the information provided in the question.

Step 1: Define the variables
Let $m$ be the number of crates of premium monitors originally ordered.
From the question, we are given that $m = 10$ .
Let $k$ be the number of crates of standard keyboards originally ordered.
Let $P$ be the cost of one crate of standard keyboards.
Since each crate of premium monitors costs double the price of a crate of standard keyboards, the cost of one crate of premium monitors is $2 P$ .

Step 2: Calculate the original total cost
The intended order consisted of 10 crates of monitors and $k$ crates of keyboards.
Original Cost $= (10 \times 2 P) + (k \times P)$
Original Cost $= 20 P + k P = P (20 + k)$

Step 3: Calculate the new (interchanged) total cost
Due to the shipping error, the quantities were interchanged. So, the delivered order contained $k$ crates of monitors and 10 crates of keyboards.
New Cost $= (k \times 2 P) + (10 \times P)$
New Cost $= 2 k P + 10 P = P (2 k + 10)$

Step 4: Use the percentage increase condition
We are given that the error resulted in a 20% increase in the total order cost.
Therefore, New Cost $= 1.20 \times$ Original Cost

Substituting the expressions for New Cost and Original Cost into the equation:

$P (2 k + 10) = 1.2 \times P (20 + k)$

Divide both sides by $P$ (since price $P > 0$ ):

$2 k + 10 = 1.2 (20 + k)$

$2 k + 10 = 24 + 1.2 k$

Subtract $1.2 k$ from both sides:

$0.8 k + 10 = 24$

Subtract 10 from both sides:

$0.8 k = 14$

$k = \frac{14}{0.8} = \frac{140}{8} = 17.5$

Step 5: Determine the initial ratio of monitor crates to keyboard crates
The initial ratio of monitor crates to keyboard crates is given by $m : k$ :

Ratio $= \frac{10}{17.5} = \frac{20}{35} = \frac{4}{7}$

Thus, the initial ratio of monitor crates to keyboard crates ordered was 4:7.