Christmas in July Math Problem

[Photo by Reenie-Just Reenie.]

In honor of my Google searchers, to demonstrate the power of bar diagrams to model ratio problems, and just because math is fun…

Eccentric Aunt Ethel leaves her Christmas tree up year ’round, but she changes the decorations for each passing season. This July, Ethel wanted a patriotic theme of flowers, ribbons, and colored lights.

When she stretched out her three light strings (100 lights each) to check the bulbs, she discovered that several were broken or burned-out. Of the lights that still worked, the ratio of red bulbs to white ones was 7:3. She had half as many good blue bulbs as red ones. But overall, she had to throw away one out of every 10 bulbs.

How many of each color light bulb did Ethel have?

Before reading further, pull out some scratch paper. How would you solve this problem? How would you teach it to a middle school student?

Do It the Hard Way: Algebra with Words

One way to approach the problem is to read each line and translate the story information into algebra. For middle school students, algebra is easier to understand if we use words as variables, rather than letters. We will use the color names to represent the number of good bulbs in each category.

…she stretched out her three light strings (100 lights each)…

That part is simple:

.

…the ratio of red bulbs to white ones was 7:3…

Be careful! Many students have trouble with ratios. The most common mistake is to write:

.

No!!

Remember that ratios are fractions. Start with the fraction relationship, and then manipulate that to get any of the following…

, which means that

,

, or

.

All of these statements are equivalent. Which one is the most useful for our problem remains to be seen.

Now That We Have the Ratio Under Control

Let’s go back and find out what else our problem has to say:

…She had half as many good blue bulbs as red ones…

Be careful again! Make sure you write down the correct relationship:

, or

.

…she had to throw away one out of every 10 bulbs…

This is another ratio, but this one is easy to figure mentally:

.

…How many of each color light did Ethel have?

.

Since the white and blue bulbs were both compared to the red, it will be easiest to put everything in terms of red bulbs. And we can put in the numbers we know for trash and total:

,

, and

.

Then we use substitution to figure out the number of white and blue bulbs.

Working with algebra leads us to some moderately complicated fraction calculations. I suspect that many middle school students would give up on the problem before reaching the end.

Is There a Better Way? Bar Diagram Ratios

Students who work with bar diagrams have little trouble with ratios. They learn that a ratio of a:b means we have a multiplication problem: there are a units of the first quantity and b units of the second quantity, with the size of the unit to be determined by the rest of the data in the problem.

…the ratio of red bulbs to white ones was 7:3…

So there are 7 units of red bulbs and 3 units of white ones:

…She had half as many good blue bulbs as red ones…

Now we add the blue bulbs to our diagram:

In order to solve our problem, we need to have all bars in terms of the same size unit. (In algebra, this would be putting everything in terms of a single variable.) Since the middle unit of the red bulbs has been cut in half by the dotted line, let’s cut all the other units in half, too. And we’ll cut the blue bar the same way:

That gives us 27 units of good bulbs. Now that we have the ratio diagram under control, we need to look at the other numbers in our problem.

…she pulled off her three light strings (100 lights each)…she had to throw away one out of every 10 bulbs…

.

As soon as we can connect a unit (or a set of same-size units) with a number, the difficult task of thinking through the problem is almost over. From here on, all we need is simple arithmetic. 27 units are 270 bulbs. That means:

.

Now that we know the size of our unit, it’s easy to find out how many bulbs of each color Aunt Ethel has.