## Algebra 1 Tutorial

#### Intro

Story problems give many students some struggles–if you’re in this group you’re definitely not alone.

We’ll explore a problem involving earning interest on savings accounts. The methods discussed here can be applied to a wide variety of problems. I’ll start my solution with a detailed step-by-step process and end with a shorter summary. Depending on your preference you may want to read the shorter summary at the end first, or the longer step-by-step process.

When you’re finished following this tutorial, you should be more confident about translating between the English and Mathematics languages when attacking story problems, as well as real world applications.

#### Sample Problem

Alice will invest \$5000 for 1 year, and wants to earn \$235 in interest (for tax reasons, she doesn’t want to earn more than that).

“Trust Us” Bank offers a savings account paying 2% simple interest. “Big Money” bank offers a savings account paying 5% interest. How much should Alice invest at each bank in order to meet her financial goal?

#### Solution

Here is a step-by-step, very detailed solution. If you prefer to start with a shorter summary of the process, just scroll down to where it says “Short Summary” towards the end.

Step 1: Understand the Problem

First of all, reread the question to make sure you understand the situation. Don’t try to do any math yet, just put yourself in Alice’s shoes for a moment and try to understand the situation.

What pieces of information are we being asked for?
What pieces of information do we know?

If you can answer these two questions, read on. If not, try once more to reread the problem with those questions in mind.

Here’s what the thought process might look like:

The big picture is that Alice is trying to set up two savings accounts and leave the money on deposit for 1 year: she wants to deposit some amount of cash at “Trust Us” Bank, and some amount of cash at “Big Money” Bank. She wants to know:

(i) how much of her \$5000 she should deposit at 2% simple interest at “Trust Us”
(ii) how much of her \$5000 she should deposit at 5% simple interest at “Big Money”

so that the combined interest equals \$235. So those are the two things we need to figure out. On the other hand, here’s some information that we know:

(A) Alice’s total investment is \$5000.
(B) The money will be in the bank for 1 year.
(C) The money earns simple interest (2% at “Trust Us”, 5% at “Big Money”).
(D) Total interest earned should be \$235.

Step 2: Translate Our Goal From English Into Mathematics

Now that we’re clear on the goal, note that (i) and (ii) above are written in English.

In order to use algebra to solve this, we need to translate the English into Mathematics.

To do this, we will assign a variable to stand for any piece of information we don’t know, but would like to know.

(i) Let stand for the amount of money Alice invests at “Trust Us”.
(ii) Let stand for the amount of money Alice invests at “Big Money”.

This gives us the building blocks to use the information from the problem in a Mathematical manner.

Step 3: Translate Known Information From English Into Mathematics

Step 3A: Information about the Account Balances

(A) Alice’s total investment is \$5000.

A sentence is to English as an equation is to Mathematics

If we want to use this information then, we have to write an equation.

English: Alice’s total investment of \$5000 is the sum of her investment at “Trust Us” and her investment at “Big Money”.

Mathematics: (Remember what and mean from step 2).

A couple of things to note here:

The English word “is” translates to the Mathematical symbol “=”.
The English word “sum” indicates two things being added together.

This alone doesn’t enable us to solve the problem. At this point, could be \$1000 and could be \$4000, or could be \$2500 and could be \$2500 (can you see why?); we just don’t have enough information to nail it down yet. To do this, we need information about the interest on the accounts too, not just the balances.

Step 3B: Information about the Interest on the Accounts

From Step 1, we had a few pieces of information about the interest on the accounts:

(B) The money will be in the bank for 1 year.
(C) The money earns simple interest (2% at “Trust Us”, 5% at “Big Money”).
(D) Total interest earned should be \$235.

Ultimately, we need a Mathematical sentence, an equation, that summarizes this information.

Let’s break this down and look at the money at “Trust Us” bank for a moment.

Recall that 2% simple interest means that if Alice deposits money at “Trust Us” for one year, then 2% of her deposit will be paid to her as interest.

The English word “of” translates to “times” in Mathematics.

So 0.02 is a Mathematical expression for the interest earned from Alice’s deposit at “Trust Us” bank.

Likewise 0.05 is a Mathematical expression for the interest earned from Alice’s deposit at “Big Money” bank.

We’re now ready to translate the English into Mathematics (see (D) above):

English: The total of the interest earned at “Trust Us” and the interest earned at “Big Money” is \$235.

Mathematics: In steps 3A and 3B, we generated two equations, two Mathematical sentences, that summarize the information in the problem. Note that we’re also looking for two pieces of information: namely the values of and .

Step 4: Finding and Steps 1-3 have all been prepwork (and these steps are where most students tend to struggle); Step 4 is where the algebra comes in. We have two equations we want to solve; I will demonstrate one method for solving them. The equations are:

(1) (2) Remember where these equations come from? Equation (1) came from step 3A and equation (2) came from step 3B. If you’re unsure how we got either equation, I encourage you to reread the relevant step above.

It’s probably simplest to solve this by the method of substitution.

Let’s get by itself in equation (1) by subtracting from both sides: Now we can substitute this expression into equation (2). Remember whenever you substitute into an equation, always put parenthesis around what you’re substituting: And we proceed to solve for as follows: (the distributive property) (combining like terms) (subtracting 250 from both sides to isolate the expression with ) (dividing both sides by -0.03 to get by itself)

In English, this means that “The amount of money Alice invests at “Trust Us” bank is \$500.” Does this make sense? If not, recall we defined in step 2. Also recall that “=” in Mathematics means “is” in English.

Now that we know , let’s substitute this into equation (1):

(1)  (substituting known value of ) (subtracting 500 from both sides to isolate )

This means that “The amount of money Alice invests at “Big Money” bank is \$4500.”

Step 5: Answering the Original Question

Many times, students can do algebra and find values, but they struggle to know when and if they are done with a story problem.

The key is: Can you answer the question posed by the problem in English?

The original question is: “How much should Alice invest at each bank in order to meet her financial goal?”

We know and . We started this process in Step 4, but let’s write our final answer by translating this back to English:

“Alice should invest \$500 at Trust Us bank and \$4500 at Big Money bank in order to earn \$235 in interest after one year.”

Since we’ve answered the original question that was posed, we know we’re done with the problem.

Short Summary:

We want to know how much Alice invests at each bank, so let’s get our goal into Mathematical language:

Let be the amount of money Alice invests at “Trust Us” Bank.
Let be the amount of money Alice invests at “Big Money” Bank.

We want to find the values of and that fit the constraints of the problem.

(1) (since Alice has \$5000 to invest in total), and
(2) (since Alice wants to earn \$235 in interest)

(See step 3 above for further explanation of these equations.)

We can solve this system of equations by substitution:

By equation (1), we have .

Substituting this into (2), we have , and we proceed to solve for as follows: (the distributive property) (combining like terms) (subtracting 250 from both sides to isolate the expression with ) (dividing both sides by -0.03 to get by itself)

Substituting into equation (1) shows us , so .

Therefore Alice should invest \$500 at “Trust Us” Bank and \$4500 at “Big Money” Bank.

Questions? Errors? Something Unclear? Something you would say differently? Drop me a line and let me know! 