Math - Working Backwards

Working backwards is a skill that can be very helpful in solving math problems with many events. It is also handy for coming up with a solution for problems with missing information that is usually presented at the beginning. 

There are two ways to do this:

Using the opposite operation when working backwards                    When you are solving a problem, try starting at the end and working backwards. Any mathematical operations you come across will need to be reversed. This means if the problem requires you to add something, then you must subtract when working backwards.

Example:

         Paul is four years older than Antonio, but Kaya is two years younger than Antonio. If Kaya is 8 years old, how old is Paul?

Solution: 
     Kaya is eight years old. Kaya is two years younger than Antonio.                             Antonio's age - 2 = 8

     So, when using the opposite operation, subtraction becomes addition                    8 + 2 = Antonio's age

     Antonio is ten years old. Paul is four years older than Antonio.                                           10 + 4 = 14

     Therefore, Paul is 14 years old.

Starting with the Answer and Working backwards                    In a problem where you know the final answer, but don't know the starting point, you can begin at the end and work your way backwards to the beginning.

Example:

Mindy baked cookies and she shared them with her family. Her father ate 3 cookies. Her mother ate 2 cookies. Her brother, Josh, ate 5. There are still 4 cookies left for Mindy. How many cookies did Mindy bake?

What do we need to find?

In this question, we need to find out the number of cookies that Mindy baked.

What do we know so far?

We know that her father ate 3 cookies, mother ate 2, and Josh ate 5 cookies. After that, there were 4 cookies left.

How do we solve this problem?

We start by indicating the total number of cookies as N. Then, we subtract the number of cookies that her family members ate. Since they were taking away cookies, we use subtraction. We equate this with the leftover 4 cookies.

Total number of cookies baked	Her father ate 3 cookies	Her mother ate 2 cookies	Josh ate 5 cookies	4 cookies were left for Mindy
N	- 3	- 2	- 5	= 4

The equation would be: N – 3 – 2 – 5  = 4

Going backward and using the opposite operation:

4 + 5 + 2 + 3 = Total number of cookies baked

14 = Total number of cookies baked

What comes next?

Going back to the original equation, we can check if our answer is correct.

14 – 3 – 2 – 5 = 4

We got the correct answer! Mindy baked 14 cookies.

Another example:

          Mrs. Garcia's class had a spelling bee and all of the students were at the front of the room together. After three minutes, five of the students had made mistakes and sat down. In the next five minutes, four more spellers sat down. One minute later, two more children sat down. In the final few minutes, one more student made a mistake, and one student was left as the winner of the spelling bee. How many children were originally in the spelling bee?

Solution:

Event		Total
> At the end, there was one winner.	1 speller	1 speller
> In the final few minutes, there was one more speller.	+ 1 speller	2 spellers
> One minute later, 2 more students sat down	+ 2 spellers	4 spellers
> In the next five minutes, 4 more spellers sat down	+ 4 spellers	8 spellers
> After three minutes, 5 students made mistakes	+ 5 spellers	13 spellers

Therefore, 13 students joined the spelling bee.

Here is another way of looking at it:

Mrs. Garcia’s class	Five students made mistakes	Four more spellers sat down	Two more children sat down	One more student made a mistake	One student was left as the winner
N	- 5	- 4	- 2	- 1	= 1

The equation would be: N – 5 – 4 – 2 – 1 = 1

Going backward and using the opposite operation:

1 + 1 + 2 + 4 + 5 = Mrs. Garcia’s class

13 = Mrs. Garcia’s class.

Did you get it? Easy right? All you have to do is practice some more and it will get even easier.

Here are more practice sheets for working backwards:

Math 2 - Working Backwards - Drill
Guided practice on using the opposite operation

Math 2 - Working Backwards - Drill
Guided practice on starting with the answer and working backwards