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Fractions Concepts

How can we change a fraction so it has a desired denominator?

Let us consider the fraction one-third drawn as a slice of cake.

The most intuitive thing to do with those big slices of cake is to keep cutting the cake.

Example 1

Make one cut so the cake has six pieces. How many are shaded?

We can represent this cake-cutting with numbers. We take ^{1}⁄_{3} and multiply the numerator and denominator by two.

Example 2

Make two cuts so the cake has nine pieces. How many are shaded?

We can also represent this cake-cutting with numbers. We take ^{1}⁄_{3} and multiply the numerator and denominator by three.

Definition

Un-Reducinga fraction represents with numbers the act of slicing the cake more. The numerator and denominator are multiplied by the same value.

Notice that there are four ways to write "un-reducing" one-third into two-sixths. Three are appropriate. One is wrong.

Example 3

"Un-reduce" the fraction

^{1}⁄_{3}into an equivalent fraction with a denominator of six.(a) write two little, floating × 2, one each for numerator and denominator

(b) write a single, centered × 2 intended to work on both numerator and denominator

(c) write another fraction: ×

^{2}⁄_{2}(d) cross out the one and three and replace them with the larger values

Do not use (b)! This is how we write multiplication, not un-reducing numerator and denominator.

We could also work backwards. We could make big slices of cake by gluing smaller slices together.

Example 4

Glue cake slices together to change two-sixths into one-third.

Example 5

Glue cake slices together to change three-ninths into one-third.

Maybe we use very sticky icing?

Definition

Reducinga fraction represents with numbers the act of gluing cake slices together. The numerator and denominator are divded by the same value.

Again there are four ways to write reducing two-sixths into one-third. Three are appropriate. One is wrong.

Example 6

Reduce the fraction

^{2}⁄_{6}into an equivalent fraction with a denominator of three.(a) write two little, floating ÷ 2, one each for numerator and denominator

(b) write a single, centered ÷ 2 intended to work on both numerator and denominator

(c) write another fraction: ÷

^{2}⁄_{2}(d) cross out the two and three and replace them with the smaller values

Do not use (b)! This is how we write division, not reducing numerator and denominator.

Notice that reducing is a process. Canceling is one possible step in the process. (Reducing and canceling are related, but not the same thing!)

How do we do fraction multiplication? Treat the situation as two independent multiplication problems. Multiply the numerators. Multiply the denominators.

The answer will be a fraction that may or may not need to be reduced.

Here is an example where the answer does not need to be reduced.

Example 7

Multiply two-thirds and seven-fifths.

Here is an example where the answer does need to be reduced.

Example 8

Multiply five-sevenths and one-tenth. (Use reducing.)

Definition

Cancelingis reducing early.

Canceling avoids big numbers when multiplying. Let's review it with an example.

Example 9

Multiply five-sevenths and one-tenth. (Use canceling.)

We cannot escape dividing both numerator and denominator by 5. But can chose to do it before or after multiplying.

Your turn. Here are three more examples. Attempt them before we do them together.

Example 10

Multiply three-fifths and one-tenth.

Example 11

Multiply six-elevenths and two-thirds.

(Try doing this problem both ways: once with reducing and once with canceling.)

Example 12

Multiply ten-fifteenths and six-eighths.

(Try doing this problem both ways: once with reducing and once with canceling.)

How do we multiply a fraction by a whole number? Remember that we can rewrite any whole number as a fraction by writing it over a denominator of one.

Chapter 2 Test, Problem 24: Simplify:

^{4}⁄_{3}× 24Chapter 2 Test, Problem 25: Simplify: 5 ×

^{3}⁄_{10}Chapter 2 Test, Problem 26: Simplify:

^{2}⁄_{3}×^{15}⁄_{4}Chapter 2 Test, Problem 27: Simplify:

^{22}⁄_{15}×^{5}⁄_{33}

How do we do fraction division? The mantra is "flip the second fraction and multiply."

Let's review it with an example.

Example 13

Divide one-half by three-quarters.

Chapter 2 Test, Problem 31: Simplify:

^{1}⁄_{5}÷^{1}⁄_{8}Chapter 2 Test, Problem 32: Simplify: 12 ÷

^{2}⁄_{3}Chapter 2 Test, Problem 33: Simplify:

^{24}⁄_{5}÷^{28}⁄_{15}

There is one other concept about fraction multiplication and division that deserves some discussion: *Why don't we need common denominators?* After all, we *do* need those when adding or subtracting fractions.

Let's do one example with pictures and commentary.

Example 14

Addone-third and one-half. (Use pictures)We start with one-third, and all the "cuts" going in one direction.

Now we want to shade in another one-half. To get halves, we "cut" in the other direction.

Now we want to fill in the top or bottom half. But part is already filled in! What should we do?

In that example we needed to "wrap around" as we filled in some of the sixths. We were forced to be aware that we are using sixths *before* looking for our final answer. This is why for addition we need a common denominator.

Now, what if we multiply one-third and one-half?

Example 15

Multiplyone-third and one-half. (Use pictures)We again start with one-third, and all the "cuts" going in one direction.

We again put the one-half in the picture by making a "cut" in the other direction.

But this time we just want one-half

ofone-third. We don't need to do any hard thinking or "wrapping around". We just re-shade half of the one-third and we're done.

For fraction multiplication we do not need to think about the new denominator until *after* doing the multiplication, when we are ready to write our answer.

Example 16

Multiply two-thirds and three-quarters. (Use pictures)

The textbook teaches a five-step problem solving process on page 54.

Familiarizeyourself with the problem situation.Translatethe problem to an equation using a variable.Solvethe equationCheckto see whether your possile solution actually fits the problem situation and is thus really a solution to the problem.Statethe answer clearly using a complete sentence and appropriate units.

To avoid the bad habit of neglecting labels until the problem is finished, we will use a slightly different six-step problem solving process:

- Determine what you are looking for
- Draw pictures
- Name things
- Make equations
- Solve the equations
- Check your answer

Each step deserves some explanation. Here is an example problem to use while discussing the steps.

Example 17

When my grandfather turned 70 he started to walk 3 miles each morning. Now he is 75 and we have no idea where he is.

Oops. That is not a math problem. Let's try again.

Example 18

When my grandfather turned 70 he started to walk around the neighborhood each morning. Below is a picture of his route. How many miles does he walk each week?

Step one is **Determine what you are looking for**. Read the problem two or three times so you understand it and notice all the details. Write down (or pretend to write) in an English sentence what you are looking for. Don't wander off track and forget what you are looking for.

Consider our example problem. We are looking for a number of miles. We need to add a bunch of numbers.

Step two is **Draw pictures**. Draw a picture or diagram of the situation. Then label things in the picture or diagram! Your visualization will not be much help without labels.

Consider our example problem. The diagram is already provided. Hooray!

Step three is **Name Things**. Write (or pretend to write) English sentences to give a one letter name to each quantity. Be aware of when two quantities are related to each other and can be expressed using the same letter. Check that each piece of information you are given is really relevant to the problem.

Consider our example problem. We are not told where on the route he starts (and stops). But that does not matter. We should just pick a corner as our "start" and label it. Let's use the top left corner.

Step four is **Make equations**. Express each relationship you know as an equation. Write (or pretend to write) an English sentence explaining each equation you write.

Consider our example problem. The sum is 1 ¼ mile + ¾ mile + 1 mile + ½ mile + ½ mile + ¼ mile

Step five is **Solve the equations**.

Consider our example problem. The two ½'s add to 1. The ¼ and ¾ also add to 1. The total is 1 mile + 1 mile + 1 mile + 1 ¼ miles = 4 ¼ miles

Step six is **Check your answer**. Check that your answer is a reasonable amount. Make sure that your final answer is in units that make sense.

Consider our example problem. An answer of four ¼ miles seems reasonable.

What can you do if you get stuck? Here are five ways to try to get unstuck.

- Imagine that you are in the problem's setting, to help use your intuition. (Maybe picture yourself tiny and in the diagram.)
- Explain to a friend in very clear English what you have done and where you are stuck.
- Look for intermediate goals: quantities to find that will help you find others.
- If the problem involves large numbers, try solving the problem with smaller numbers instead. Once you see the steps to use, go back and do those steps with the original numbers.
- If the problem is complicated, try simplifying it and solving the simpler problem, to get ideas on how to approach the original problem.

For most people, the hardest part of a word problem is drawing the picture, naming the amounts, and especially setting up the equation. We will have a lot of practice doing these steps with simple word problems. Your textbook has pages called *Translating for Success* which allow you to practice only these beginning steps.

Also remember which English words and phrases correspond with which arithmetic operations. Addition is usually "sum", "total", "increased by", or "more than". Subtraction is usually "difference", "how much more", "decreased by", or "less than". Multiplication is usually "of" or "times". Division usually lacks a phrase but is about finding equal portions.