Using the Separation Method to Solve Math Problems

Instructor: Babita Kuruvilla

Babita has an electrical engineering degree and has taught engineering students and college students preparing for medical and dental college admissions tests.

In this lesson, we use the separation method to work through math problems. We explore the three different types of separation problems. We'll also define the word variable and discuss its use in a mathematical expression.

Mathematical Building Blocks

Imagine that you are given some random alphabet blocks and asked to make a word. Depending on the order in which those letters are arranged, you'd probably come up with a few different words.

Alphabet blocks
alphabet blocks

What if I were to give you building blocks with numbers and mathematical signs written on them instead of letters? Could you build a meaningful expression using those blocks? That answer should be a resounding ''Yes!!''

We can write mathematical expressions such as 2 + 3 = 5 and 7 - 4 = 3. Do you know what those mean?

Symbols and Variables in Expressions

The first expression 2 + 3 = 5 is a simple addition expression. It tells us that if we take two of anything and add three more, we then have five altogether. On the hand, the second expression 5 - 2 = 3 says that if there were five of something and we took away two, then we have three left. This is a separation expression.

Sometimes, we can add symbols or letters to an expression. Take a look at x + 3 = 8. In this equation, x is a variable. It is the part of the equation that we don't know yet. A constant in an expression has a fixed value. In this case, 3 and 8 are both constants. By using some math manipulation, we can find the value that x has for this equation.

Let's say that you had a basket of fruits. You sort them out and find that you have 7 apples and 8 peaches. How could you write this in mathematical form? You can add 7 and 8 to get 15 pieces of fruit, but how do you say that 7 are apples and 8 peaches? We can use symbols or letters to represent each kind of fruit. If we use A for apples and P for peaches, then, we can write the total number of fruits, T = 7A + 8P = 15. In this case, we used letters to represent the different types of fruits. They did not have an assigned value, they simply represent a type of fruit.

When we write mathematical expressions using variables, in addition to representing specific items, they have a value. So, variables are a great way to write expressions when we need to figure out one part of the equation that we don't know yet.

Using the Separation Method

Often, we are given real-life situations in word problems. Let's say George goes to the market and buys 7 oranges. If he gives 2 oranges to Sarah, then he has 5 left. The mathematical expression is 7 - 2 = 5.

There are three different values given in that scenario:

  • What George initially had - Initial quantity (I)
  • What he ate - Change in quantity (C)
  • What he has left - Final quantity (F)

Our equation is I - C = F.

If two of these values are known, we can always find the third unknown value. So, whenever you are working on word problems, the first step is to always write out the given values and then identify the unknown.

Let's look at three different separation type problems.

Initial Quantity Unknown

Example 1 - George goes to the market and buys some oranges. If he gives 2 oranges to Sarah, he has 5 left. How many oranges did he buy at the market?

The givens are the change in quantity (C) and the final quantity (F). The unknown is the initial quantity (I). So, we write out the expression (I - C = F) plugging in the known and unknown values.

Since George gave away 2 oranges, this is a separation problem. That is, two oranges were separated from the initial quantity. So, our expression will be:

I - 2 = 5

Solve for the initial quantity by adding the 2 to both sides so we have the variable on a side by itself:

I - 2 + 2 = 5 + 2

I = 7

Remember that in math it is legal to something to one side if you do the same thing to the other side.

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