Question

Rob is a high school student taking both algebra and geometry. Last night he had 27 homework problems combined from the two classes, and he took a total of 90 minutes to do them. If each algebra problem took him 2 minutes and each geometry problem took him 5 minutes, how many algebra problems did he have for homework?
2
5
12
15

Answer 1

Let \( a \) be the number of algebra problems and \( g \) be the number of geometry problems. We know the following two equations based on the information provided:

1. The total number of problems:
\[
a + g = 27
\]

2. The total time spent on the problems:
\[
2a + 5g = 90
\]

Now we can solve this system of equations. First, we will express \( g \) in terms of \( a \) using the first equation:

\[
g = 27 - a
\]

Next, we can substitute this expression for \( g \) into the second equation:

\[
2a + 5(27 - a) = 90
\]

Now, distribute the \( 5 \):

\[
2a + 135 - 5a = 90
\]

Combine like terms:

\[
-3a + 135 = 90
\]

Now, isolate \( a \):

\[
-3a = 90 - 135
\]
\[
-3a = -45
\]
\[
a = 15
\]

Thus, Rob had \( 15 \) algebra problems for homework.

Now let's verify this by calculating \( g \):

Using \( g = 27 - a \):
\[
g = 27 - 15 = 12
\]

Next, we check the total time:

\[
2a + 5g = 2(15) + 5(12) = 30 + 60 = 90
\]

Since both the number of problems and the total time are correct, we conclude that the number of algebra problems Rob had for homework is \(\boxed{15}\).