## Working Backwards

### What is the Working Backward Method?

The Working Backwards Method is a Singapore Math problem-solving strategy that’s introduced in P3 Math classes. It is usually used to solve Math questions that involve a series of related events.

### How does the Working Backwards Method work?

Think about a time when you misplaced something. What did you do?

Did you retrace your steps and think of the places you have been to before getting where you are now?

If you had walk from home to the parade square, from the parade school to class, and then from your class to the canteen, the natural thing to do to start your search would be to move in the reverse order starting from the end point (the canteen) back to your class, then from your class to the parade square, and from the parade square back home. In doing so, you’re bound to find your missing item lying somewhere.

Well, this act of retracing is a form of working backward. If you understand this, you’ll know how the Working Backwards method works!

### Examples of Working Backwards Math Problems

Let’s look at some questions that require us to use the Working Backwards method!

Spongey boarded the bus when there were some passengers on it. When the bus arrived at the next stop, an additional 8 people boarded the bus and 3 passengers alighted. 2 stops later, 11 people boarded the bus. All 24 people alighted the bus at the bus interchange. How many passengers were there on the bus when Spongey boarded it?

Brad and Bruce had some Lego bricks each. Brad gave 1/6 of the Lego bricks he had to Bruce. Bruce then gave 2/3 of the Lego bricks he had to Brad. In the end, Brad had 270 Lego bricks and Bruce had 45 Lego bricks. How many Lego bricks did Bruce have at first?

### How do we identify questions that need us to work backwards?

Looking at the examples above, what did you notice about the questions?

Questions that can be solved by Working Backwards can be divided into 3 main stages:

1. The beginning: There is always important missing information at the start.
2. A Series of Events takes place
3. The end: This is where a final number is given to us.

Using the first question above as an example, notice how it starts by telling us there were “some passengers”, not telling us the exact number of people there were?

Next, 8 people got on, 3 people alighted, 11 boarded. These events are linked and happen one after another and we know the exact number of people involved during each event.

Lastly, we know what the final number of passenger is.

### How do we master the Working Backwards Method?

Retracing the steps and undoing each stage helps us solve the missing pieces in between to find that missing information. The main idea is to solve the problem by reversing the steps from the back to get back to the beginning.

In order to do that, there are 2 important things that we’ll need to know:

1. Identify the number of events that have occurred
2. Undo/Reverse the effect of each event

Let’s use the first question as an example.

Spongey boarded the bus when there were some passengers on it. When the bus arrived at the next stop, an additional 8 people boarded the bus and 3 passengers alighted. 2 stops later, 11 people boarded the bus. All 24 people alighted the bus at the bus interchange. How many passengers were there on the bus when Spongey boarded it?

#### A. Identifying the Number of Events that have occurred

After reading the question, we know that the problem takes place as a bus travels from some place to the bus interchange. So let’s try to break the problem up into smaller chunks and look at each sentence to divide them into stages.

In the beginning, Spongey boarded the bus and we do not know the existing number of passengers on the bus.

Then, event #1 happens where we have people getting up and down the bus.
As 8 people got up the bus and 3 people got down, the original number of passengers on the bus changes.

Next, event #2 happens. Some people got on the bus at another stop and this changes the new no. of passengers that we had in the previous stage.
After 11 people got on the bus, the newer no. of passengers on the bus can be found by adding 11 people to the new number of passengers in the last stage.

In the end, we are told that the total number of passengers on the bus is 24 at the bus interchange. In other words, the newer no. of passengers was 24.

#### B. Time to Undo the Done

See how we started off not knowing the original number of passengers on the bus? However, as the bus travels, we know that there are 2 main events which occurred that changes the total number of passengers on the bus each time. In addition, we are given the total number of passengers in the end.

Now, it’s time to solve the puzzle by undoing the effects of each event to find out the original number of passengers on the bus!

Now, starting at the end of the problem, let’s try to undo each event one step at a time until we reach the beginning of the question.

The main idea is to identify the operation ( + / – / x / ÷ ) and then reverse it to get to the previous stage.

PRO TIP: When you think of Opposites, think of ADDING vs SUBTRACTING, MULTIPLYING vs DIVIDING.

Since we are aware that there were 24 passengers on the bus in the end, this is where we’ll start.

In order to reverse the situation such that Event #2 has not happened, we will need to take away the 11 passengers who have gotten on the bus during Event #2.

24 – 11 = 13

So now we know, we have 13 passengers on the bus before Event #2 happened.

So far so good?

Now, how do we reverse the situation such that Event #1 has not happened?

To get 13 passengers, we added 8 people and subtracted 3 people from the beginning. Hence, to undo that, we’ll need to subtract 8 people and add 3 people from 13 so that we can get the original number of passengers in the beginning.

13 – 8 + 3 = 8

Congrats! We have now solved the question and found that there were 8 passengers at the start when Spongey first boarded the bus.