Finding a total when the fractional part is known

Someone asked me recently,

I have had great success with the Grade 4 Light Blue Series with my twin daughters and we have been breezing through the curriculum. ... For the most part, they get how to find the fractional part but we did have some trouble on problems 4 on page 59 Worktext 4-B and problem 7 on page 60 in Worktext 4-B.

They have to do with finding the total number of something when you are given a fractional part.

I had a hard time teaching the concept of "working backwards" and was wondering if you have any additional resources or problems we could use to practice.

Working backwards is of course perfectly FINE but I feel using the BAR MODEL is easier for children who have not worked through these kinds of problems before. The model shows them clearly HOW we "work backwards" to get the total.

Here's an example from the previous page (page 58 in Math Mammoth 4-B):

This is the problem #4 they had trouble with

One cold day, 1/8 of a company's workers were sick and stayed home. 84 people showed up at work. How many workers does the company have?

You can simply teach the children to make a bar model for these types of problems. Then once they get used to that, they may also start solving them without the model, but the model is the starting point so they can see clearly the relationship between the fractional parts and the total.

For this problem, draw a bar divided into 8 parts. Mark on it the 84 people (which is seven  parts).

←------------  ? --------------→
|---|---|---|---|---|---|---|---|
←------------ 84 ----------→

Mark the whole bar with a question mark so it is clear THAT is the unknown.

Then the solution should be easy to see! The 84 people correspond to seven parts in the model. Find how much one part is: 84 ÷ 7 = 12. Then, the total is eight of those parts: 8 x 12 = 96.

Now try this one. Let's change 1/8 into 5/8 in the above problem: One cold day, 5/8 of a company's workers were sick and stayed home. 84 people showed up at work. How many workers does the company have?

This time you mark three of the eight 'blocks' as being 84:

←------------- ? ---------------→
|---|---|---|---|---|---|---|---|
←--- 84 ---→

The solution: first divide 84 ÷ 3 = 28 to find the value of one "block". Then, multiply that by 8!

This topic comes up later in the curriculum, too. For example, the lesson Problem Solving with Bar Models 4 in Math Mammoth Grade 5-A concentrates on these types of problems.

I don't currently have any additional worksheets that would concentrate on these types of problems but you can make questions like these to practice the basic idea:

2/9 of a number is 66. What is the number?
5/7 of a number is 35. What is the number?

Hope this helps!
~Maria