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.
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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.
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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
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Her
father ate 3 cookies
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Her
mother ate 2 cookies
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Josh
ate 5 cookies
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4
cookies were left for Mindy
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N
|
-
3
|
-
2
|
-
5
|
=
4
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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
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Total
|
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> At the end, there was one winner.
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1
speller
|
1
speller
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> In the final few minutes, there was one
more speller.
|
+
1 speller
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2
spellers
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> One minute later, 2 more students sat
down
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+
2 spellers
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4
spellers
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> In the next five minutes, 4 more spellers
sat down
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+
4 spellers
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8
spellers
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> After three minutes, 5 students made
mistakes
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+
5 spellers
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13
spellers
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Therefore, 13 students joined the spelling bee.
Here is another way of looking at it:
Mrs.
Garcia’s class
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Five
students made mistakes
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Four
more spellers sat down
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Two
more children sat down
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One
more student made a mistake
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One
student was left as the winner
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N
|
- 5
|
- 4
|
-
2
|
-
1
|
=
1
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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
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
