|Math 20 Math 25 Math Tips davidvs.net|
Why do we want to change denominators? The most common motivation is needing to add fractions.
The denominator of a fraction acts like a word describing the type of thing we are considering. The fraction 3/4 says "I am considering fourths, and care about three of them." This is very similar to the phrases "3 inches" or "3 centimeters".
Because of how denominators are conceptually like labels, trying to add fractions with unlike denominators works as badly as trying to add inches and centimeters. We need to change the fractions so their numerators are counting the same type of thing. We need to make their denominators match.
Consider the cake below. Amy gets to eat the big shaded half. Beatrice gets to eat the smaller shaded fourth. How much of the cake have they eaten?
When we change the denominators of two fractions so they match, there are three situations.
Sometimes one denominator is a multiple of the other.
Change one-twelfth and one-sixth to have common denominators.
Sometimes there are no common factors.
Change one-fourteenth and one-fifteenth to have common denominators.
Sometimes a few factors are common, but not all for either denominator.
Change one-twelvth and one-fifteenth to have common denominators.
The least common denominator of two or more fractions is the smallest denominator that can be added.
We could use a bigger one. But we would needlessly duplicate common factors, and would always have to reduce the answer. (We'll see this happen soon.)
If we use the least common denominator we might have to reduce our answer. But it is no longer a step that will always be needed.
Two or more fractions are relatively prime if their least common denominator is the product of their denominators.
In other words, denominators are relatively prime we cannot find any common factors.
Let's do the same fraction addition problem in three different ways. How should we add 1/30 + 1/42?
The first method we'll nickname the brute force method. We find a new denominator very quickly be multiplying each old denominator by the other. This always works, but forces us to deal with large numbers.
Method One: Brute Force
Add 1/30 + 1/42.
Start by multiplying each denominator by the other.
The second method we'll nickname the factor tree method. Using a factor tree with each denominator we can be absolutely sure to avoid needlessly duplicating factors. We make the denominators match by making the factors of the new identical denominators match, putting in "missing" factors from each but avoiding unneccesary duplicates.
Method Two: Factor Trees
Add 1/30 + 1/42
Start by recognizing that the new denominators will match as soon as they match as products of factors.
The third method is to make a list of multiples for each denominator. The smallest number on both lists will be the optimal denominator. This fourth method is slow, but is favored by some students who prefer to work with multiples instead of factors.
Method Three: Lists of Multiples
Add 1/30 + 1/42
Start with the insight that the common denominator must be a multiple of each of the original denominators.
Multiples of 30: 30, 60, 90, 120, 150, 180, 210, 240
Multiples of 42: 42, 84, 126, 168, 210, 252
Time for a few more examples. You try these before seeing the instructor's work. Use any of the three methods.
Add 1/36 + 1/6
Subtract 3/4 − 1/12
Add 1/5 + 1/4
Add 1/12 + 2/15
Subtract 5/42 − 1/24
Chapter 3 Test, Problem 3: Simplify: 1/2 + 5/2
Chapter 3 Test, Problem 4: Simplify: 7/8 + 2/3
Chapter 3 Test, Problem 6: Simplify: 5/6 − 3/6
Chapter 3 Test, Problem 7: Simplify: 5/6 − 3/4
Chapter 3 Test, Problem 8: Simplify: 17/24 − 1/15
Note that the same techniques work when we have more than two fractions. The problem just takes longer because it is more work to find which factors are missing from among more than two numbers.
Add 1/4 + 1/6 + 1/15
Chapter 3 Test, Problem 5: Simplify: 7/10 + 19/100 + 31/1,000
Finally, remember that we can cancel factors but we cannot cancel terms.
There are two ways to think about subtracting with mixed numbers.
We could treat the fractions as a place value column and borrow from the 1's column if we need to do so.
We could change both mixed numbers to improper fractions and then subtract.
Let's do the same pair of examples using both methods.
You cut 2 1/2 inches off a 3 3/4 inch board. How long is the remaining part? (Think like place value.)
You cut 2 1/2 inches off a 3 3/4 inch board. How long is the remaining part? (Use improper fractions.)
You cut 2 1/2 inches off a 3 1/4 inch board. How long is the remaining part? (Think like place value.)
You cut 2 1/2 inches off a 3 1/4 inch board. How long is the remaining part? (Use improper fractions.)
It is helpful to be fluent with both these methods of subtracting mixed numbers. For some problems the first will be easier. For other problems the second will be easier.
Chapter 3 Test, Problem 19: Simplify: 10 1/6 − 5 7/8
To change a fraction into a decimal, we just treat the fraction as a division problem. We do numerator ÷ denominator = decimal
1/5 = 1 ÷ 5 = ?
3/8 = 3 ÷ 8 = ?
There is a shortcut for changing a fraction to a decimal if the denominator is a factor of 10, 100, etc.
If we "un-reduce" to make an equivalent fraction whose denominator is 10, 100, etc. then we can simply read the new fraction.
1/25 = 1/25 × 4/4 =
With practice, you wll be able to look at a decimal and see it in several ways, simultaneously.
Consider the decimal 0.2 which has:
(Do not worry now about the percent name. We will study percents later.)
To change a normal decimal into a fraction, we just say its formal name and then simplify. This is one reason why we need to remember place value names!
Write 0.08 as a fraction.
Remember that there are four ways to write reducing eight hundredths into two twenty-fifths. Three are appropriate. One is wrong.
Write 0.08 as a fraction.
(a) show reducing by writing two little, floating ÷ 4, one each for numerator and denominator
(b) show reducing by writing a single, centered ÷ 4 intended to work on both numerator and denominator
(c) show reducing with a fraction: ÷ 4/4
(d) show reducing by crossing out the eight and twenty-five and replacing them with the reduced values
Do not use (b)! This is how we write division, not reducing numerator and denominator.
Write 0.375 as a fraction.
The most fun way to review converting decimals to fractions is to play The Decimals to Fractions Game.
Game Rules: The Decimals to Fractions Game
A moderator provides five numbers.
Students, in teams with one representative at the board, race to change three of the numbers at a time nto a correct decimal-to-fraction equation. The three numbers used become the decimal, numerator, and denominator.
How many solutions can the class find?
The five numbers are 1, 2, 5, 6, and 30.
One possible solution uses the three numbers 1, 2, and 5 to create the equation 0.5 = 1/2.
A second possible solution uses the three numbers 2, 6, and 30 to create the equation 0.2 = 6/30.
The five numbers are 1, 4, 8, 25, and 125.
The five numbers are 4, 8, 16, 20, and 25.
The five numbers are 3, 6, 15, 20, and 75.
An .svg image with both the instructions and the outline is here.
Repeating decimals never end and have a pattern. We write them using a bar over the repeating portion instead of ellipsis to avoid ambiguity.
Which do we mean by 0.321...
0.3211111...? 0.321212121... 0.321321321...
Note that a repeating decimal has no formal name. So we cannot use the usual method to change it into a fraction.
For the three most common repeating decimals it is best to simply memorize their fraction equivalents.
One-third = ?
Two-thirds = ?
One-sixth = ?
There is a method for changing a repeating decimal into a fraction, but it is tricky.
Knowing how to do this is not required for Math 20 students. This tricky method will not appear on any homework assignment, quiz or test!
The method involves setting the repeating decimal equal to y, multiplying both sides of the equation by a 10 one or more times, and then subtracting the two equations before solving.
It makes more sense when you see it happen.
Change 0.222222... into a fraction.