Now, let’s solve some interesting word problems with our skills. Word problems require you to interpret the presented information mathematically.
We’ll try the following example and let you try some more. No peeking at the explanation until you’ve attempted it!
Example 3:
Sid has \( 5 \) more than three times the amount of money in his savings account in July than he did in June. The sum of his balance in June and July is \( \$140 \). How much money does he have in July?
Hmm… this seems like an interesting problem.
Let’s first note what we want to find. We want to find the amount of money Sid has in July.
Here, note that we can’t solve this problem quickly in our heads or with trial-and-error quickly. To see why, let’s try:
Suppose Sid has fifteen dollars in June – let’s see if this is valid according to our problem statement. If it is, we can solve for the correct amount of July. Since Sid has “\( 5 \) more than three times” the amount he had in June (fifteen dollars), he has \( 5 + 3 \times 15 =50 \) dollars in July. The sum of his balance in June and July is thus \( 50 + 15 = 65 \) dollars. Oops… that’s not \( \$ 140 \) as we were told.
We could try another number for Sid’s balance in July and guess our way to the right answer, but typically, a problem that seems like a lot of trial and error usually calls for our algebra tools.
Since there are a lot of quantities we don’t know, let us identify our unknowns. We don’t know how much he had in June. We don’t know our answer, the amount he has in July, which is based off of the amount he had in June.
Hmm… his amount in July is based off the amount he had in June? How so?
Reading the first sentence in our statement tells us:
Sid has \( 5 \) more than three times the amount of money in his savings account in July than he did in June.
When we did our trial and error, we noticed that we had to use this piece of information to obtain the amount he had in July from the amount he had in June. Say he had \( x \) dollars in June. What’s the amount he has in July?
Five more than three times the amount of money he had in June.
Let’s break this down. Three times the amount of money he had in June is simply \( 3 \times x \), or \( 3x \). Five more than “three times the amount of money he had in June” is five more than this quantity, or \( 5 + 3x \). Thus he has \( 5 + 3x \) dollars in July.
Even though our quantities are expressed in terms of a variable, we still haven’t solved the problem. Luckily, our second sentence provides a saving grace:
The sum of his balance in June and July is \( \$140 \).
Oh, now we see our equation!
He had \( x \) dollars in June and \( 5+3x \) dollars in July, which we figured out above by letting the amount we had in June be a variable \( x \). These sum to 140 dollars. As long as we solve for this variable, we can figure out our answer, which is \( 5 + 3 \cdot x \).
Hence, our equation is:
\[ x+(5+3x)=140 \]
Now we can solve this equation like any old linear equation – do operations on both sides, moving variables on constants onto separate sides, isolating \( x \), etc. Solving gives us
\[ 4x+5=140 \]
\[ 4x=135 \]
\[ x= 33.75 \]
In June, Sid had \( \$ 33.75 \). In July, he has \( 5+3x=5+3(33.75)=106.25 \) dollars, which is our answer.
Oftentimes the hardest part about word problems is choosing which one of your unknowns be variable and interpreting the information presented mathematically.
Try these word problems on your own. Answers are on the next page:
- John has $4 more than twice the amount of money in his wallet compared to what he had yesterday. The sum of the amounts he had yesterday and today is $64. How much money does he have today?
- Sarah has $3 less than four times the amount of money in her piggy bank compared to last month. The total amount of money in her piggy bank last month and this month is $77. How much money does she have now?
- On a farm, there are pigs and chickens. The total number of animals is 24, and the total number of legs is 76. How many pigs and how many chickens are on the farm?