Question

Andrea is buying some new shirts and sweaters. She is able to buy 3 shirts and 2 sweaters for
, or she is able to buy 2 shirts and 4 sweaters for
. How much does the item with the lowest value cost?


16


24


4


8

Answer 1

Let's denote the cost of a shirt as \( S \) and the cost of a sweater as \( W \).

From the problem statement, we can set up the following equations based on the combinations of shirts and sweaters Andrea can buy:

1. For 3 shirts and 2 sweaters:
\[
3S + 2W = 16
\]

2. For 2 shirts and 4 sweaters:
\[
2S + 4W = 24
\]

Now, we solve these equations step by step.

First, we can simplify the second equation:
\[
2S + 4W = 24 \implies S + 2W = 12 \quad \text{(divide both sides by 2)}
\]

Now we have the two equations:
1. \( 3S + 2W = 16 \)
2. \( S + 2W = 12 \)

Next, we can solve for \( S \) in terms of \( W \) using the second equation:
\[
S = 12 - 2W
\]

Now we'll substitute \( S \) into the first equation:
\[
3(12 - 2W) + 2W = 16
\]
Expanding this gives:
\[
36 - 6W + 2W = 16
\]
Combining like terms:
\[
36 - 4W = 16
\]
Subtracting 36 from both sides:
\[
-4W = 16 - 36 \implies -4W = -20
\]
Dividing both sides by -4:
\[
W = 5
\]

Now substitute \( W = 5 \) back into the equation \( S + 2W = 12 \):
\[
S + 2(5) = 12
\]
\[
S + 10 = 12
\]
Subtracting 10 from both sides:
\[
S = 2
\]

Thus, we have:
- The cost of a shirt (S) is \( 2 \).
- The cost of a sweater (W) is \( 5 \).

The item with the lowest value is the shirt, costing \( 2 \).

From the given options:
- 16
- 24
- 4
- 8

None of these options directly match the cost of the shirt or sweater. If we instead need to find the item with the lowest value from the provided options without a match in our derived values, we should pick the lowest number, which would be **4**.

Thus, the answer is **4**.