Question

An employee at a party store is assembling balloon bouquets. For a graduation party, he assembled 6 small balloon bouquets and 3 large balloon bouquets, which used a total of 126 balloons. Then, for a Father's Day celebration, he used 246 balloons to assemble 7 small balloon bouquets and 8 large balloon bouquets. How many balloons are in each bouquet?

Answer 1

Let \( s \) represent the number of balloons in a small bouquet, and \( l \) represent the number of balloons in a large bouquet. We can set up the following system of equations based on the information provided:

1. From the graduation party, we have:
\[
6s + 3l = 126
\]

2. From the Father's Day celebration, we have:
\[
7s + 8l = 246
\]

To solve this system of equations, we can first simplify the first equation by dividing everything by 3:
\[
2s + l = 42
\]

We can express \( l \) in terms of \( s \) from this simplified equation:
\[
l = 42 - 2s
\]

Next, we substitute \( l \) into the second equation:
\[
7s + 8(42 - 2s) = 246
\]
Expanding this, we have:
\[
7s + 336 - 16s = 246
\]
Combining like terms gives:
\[
-9s + 336 = 246
\]
Now, subtract 336 from both sides:
\[
-9s = 246 - 336
\]
\[
-9s = -90
\]
Dividing both sides by -9, we find:
\[
s = 10
\]

Now that we have \( s \), we can substitute it back into the equation for \( l \):
\[
l = 42 - 2(10) = 42 - 20 = 22
\]

Now we have the number of balloons in each type of bouquet:
- Small bouquet: \( s = 10 \)
- Large bouquet: \( l = 22 \)

Finally, let's verify our solution by substituting these values back into the original equations.

For the first equation:
\[
6(10) + 3(22) = 60 + 66 = 126 \quad \text{(correct)}
\]

For the second equation:
\[
7(10) + 8(22) = 70 + 176 = 246 \quad \text{(correct)}
\]

Thus, the solution is confirmed. Therefore, the number of balloons in each bouquet is:
\[
\boxed{10} \text{ (small bouquet)}, \quad \boxed{22} \text{ (large bouquet)}
\]