CHALLENGE 4: Add these fractions and reduce to lowest terms:

1/6 + 2/6 = ? , 3/8 + 3/8 = ? , 2/9 + 4/9 = ?

EXAMPLE 3: Kevin wants 1/2 of a pizza pie and Karen wants 1/4 of a pie? Draw a picture of a pizza pie. Divide it into halves. Divide one of the halves into quarters.
The total of what they want is what fraction of a whole pie?

THINK: What is the number question?
It is 1/2 + 1/4 = ?
By looking at the pie can you figure out what the sum is? Can you figure out the rule for adding fractions with different denominators?
COMPUTE: Let's look at common fraction model (c). Look at the denominators of the two fractions. You can't add the numerators unless the denominators are the same. Remember, you can add oranges and oranges but not oranges and grapefruit. So, you must change the fractions so that they have the same denominator, which we call a common denominator. You can always find a common denominator by multiplying one denominator by the other, 2 x 4 = 8, but if you find the lowest common denominator or LCD, you won't need to reduce the final fraction so much. In this case 8 is not the LCD. You can find the LCD by multiplying each denominator by 2, 3, 4, etc. to find a product that is the same for both. That is the common denominator.

1. Start with the 1/2. Multiply the denominator by 2. We don't need to go any further because 4 is the denominator of the other fraction.
   
2. When you multiply the denominator by 2, what must you do to the numerator? So instead of 1/2 you now have 2/4.
   
3. We don't need to do anything to the 1/4 because its denominator is the common denominator.
   
4. Now we can add the two fractions by adding the numerators.

CHALLENGE 5: Paul ate 1/6 of a candy bar. His brother ate 1/3 of the bar. Between them what fraction of the candy bar did they eat? Reduce the fraction in your answer to lowest terms.

CHALLENGE 6: Think of addition story problems that can be expressed by each of these number questions, and find the sums:

1/3 + 1/6 = ? , 1/4 + 1/6 = ? , 3/8 + 1/4 = ?

EXAMPLE 4: There was 1/2 of a whole pizza in the refrigerator. It was cut into 2 equal pieces. Kyle ate one of the pieces. What fraction of the whole pizza was left?

THINK: Draw a sketch of a pizza and cut it into 2 pieces. Cut one piece into two pieces. Shade one piece. What fraction of the whole is the shaded piece? Is the problem a combining problem or a separating problem? What is the number question? Is it 1/2 - 1/4 = ?
COMPUTE: Look at common fraction model (d) for subtracting fractions. Again we must find a common denominator, which is 4, and we change 1/2 to 2/4. We subtract one numerator from the other, 2 - 1 = 1. So, 1/4 is left.
CHECK: Look at the drawing to see what fraction is left.

CHALLENGE 7: Mrs. Goodcook, had 2/3 of a cup of milk. She used 1/3 of a cup of milk in her gravy. How much milk did she have left?

CHALLENGE 8: Mrs. Jones picked 3/4 of a quart of strawberries. Her boys ate 1/8 of a quart. What fraction of a quart was left?

CHALLENGE 9: Make these subtractions:

5/8 - 3/8 = ? , 7/8 - 1/4 = ?, 7/10 - 1/5 = ?

EXAMPLE 5: Suppose you pour 1-3/4 (one and three-quarter) cups of water into a four cup measuring pitcher. Then you measure 1/2 cup of water in a one-cup measuring cup and pour it into the pitcher. What will the total be in the pitcher?

The 1-3/4 is called a mixed number because it consists of a whole number and a fraction. The number question is 1-3/4 + 1/2 = ?

THINK: Is the number question 1-3/4 + 1/2 = ?
COMPUTE:

1. Hold aside the 1 in 1-3/4 and just add the two fractions: 3/4 + 1/2 = ? Find the common denominator, which is 4, change 1/2 to 2/4, and make the addition on paper: 3/4 + 2/4 = 5/4
   
   • Change the improper fraction (5/4) to a mixed number. You can subtract 1 (4/4) from 5/4:
     5/4 - 4/4 (1) = 1/4. So 4/4 (1)+ 1/4 = 1-1/4
     
2. Now we go back and pick up the 1 that we set aside. On paper add the 1 to the sum of the fractions.
   1 + 1-1/4 = 2-1/4.

[Another way to make the addition is to change the mixed number, 1-3/4, into an improper fraction, meaning that the numerator is larger than the denominator and then add the fractions: (1) Change the 1 to a fraction with a denominator of 4. That fraction is 4/4. (2) Add the two fractions: 4/4 + 3/4 = 7/4. Now you can add 7/4 + 1/2 = 7/4 + 2/4 = 9/4 = 2-1/4

There is another way to change a mixed number to an improper fraction that you may have learned. Multiply the whole number by the denominator of the fraction (1 x 4 = 4). Then add the product (4) to the numerator of the fraction (4 + 3 = 7), which is the numerator of the improper fraction (7/4).]

CHALLENGE 10: Mrs. Brown had 2-1/2 cups of broth. She added 1-1/4 cups of water to the broth. Then how much liquid did she have? Solve by adding the two fractions and adding the sum to the sum of the whole numbers.
Back to the Beginning

EXAMPLE 1: Mrs. Jones's recipe calls for 1/4 cup of nuts for a dozen cookies. She wants to make 1/2 dozen cookies. What fraction of a cup of nuts does she need?

THINK: She wants to make 1/2 of the recipe, so she needs 1/2 of 1/4 cup of nuts. The number question is 1/2 x 1/4 = ?
When we are dealing with fractions and see the word "of," it usually means "times."
COMPUTE: Let's look at common fraction model (e) for multiplying fractions. Can you figure out what the rule is for multiplying fractions? Take you time.
The rule in multiplying fractions is that we multiply one numerator by the other and one denominator by the other. Then we reduce the fraction to its lowest terms, but in this case it is already reduced to it lowest terms.
CHECK: Look at a measuring cup or draw a picture of one. You can see that 1/2 of 1/4 is 1/8.

CHALLENGE 1: Mrs. Jones also used 1/8 of a 1/2 pound package of butter. How much butter did she use?

CHALLENGE 2: Harry bought 1/2 of a pizza pie. He could only eat 2/3 of the 1/2 piece. He ate what fraction of the whole pizza? Reduce the fraction to lowest terms.

CHALLENGE 3: Mary gave Kevin 1/2 of her candy bar. Kevin gave Richie 1/2 of his 1/2 piece. What fraction of the whole bar did Richie get? Draw a picture showing how the bar was divided to help you understand the problem.

CHALLENGE 4: Think of a multiplication story problem that can be expressed by this number question, and find the product: 1/2 x 1/3 = ?

CHALLENGE 5: Make these multiplications of fractions:

2/3 x 1/4 = ? , 3/4 x 2/3 = ? , 1/3 x 3/8 = ?

EXAMPLE 2: Sue has 1/2 pound of chocolates that she wants to split into 1/4 pound bags. How many 1/4 pound bags of chocolate can she make?

THINK: The problem is how many 1/4s can she separate from 1/2. Is the number question 1/2 ÷ 1/4 = ?
COMPUTE: Let's look at common fraction model (f) for dividing fractions. The rule is that to divide fractions, you invert the divisor and multiply and get 4/2, which can be reduced to 2.
CHECK: Multiply the divisor, 1/4, by 2, and you should get the dividend, 1/2.

CHALLENGE 6: Mrs. Goodcook had 3/4 cup of molasses. She needs 1/8 cup for a dozen cookies. How many dozen cookies can she make with the molasses?

EXAMPLE 3: Granny wants to make a meat loaf, and her recipe calls for 5-1/2 pounds of ground beef. She only wants to make 1/3 of the recipe. How many pounds of beef does she need?

THINK: Do you need to multiply or divide? What is the number question?
COMPUTE: You need to multiply 1/3 times 5-1/2, which is a mixed number.

1. You need to change the mixed number to an improper fraction. Do you remember how to do that?
   
   • Change 5 to an improper fraction with a denominator of 2 because 1/2 has a denominator of 2. So 5 x 2/2 = 10/2.
     
   • 10/2 + 1/2 = 11/2.
     
2. 1/3 x 11/2 = 1 x 11 / 3 x 2 = 11/6
   
3. Change the improper fraction to a mixed number: 11/6 = 6/6 + 5/6 or 1-5/6

CHECK: 5-1/2 x 1/3 = 1-5/6 To check: 1-5/6 ÷ 1/3 should equal 5-1/2. So we invert 1/3 and multiply. 1-5/6 x 3 = 11/6 x 3 = 33/6 = 5-3/6 or 5-1/2. It checks.

CHALLENGE 7: Timmy has 3-3/4 gallons of lemonade to sell. Fran has only 1/3 as much. How much lemonade does Fran have to sell?

CHALLENGE 8: Do these divisions:

1/2 ÷ 1/4 = ? , 3/4 ÷ 1/8 = ? , 6/10 ÷ 1/5 = ?

Back to the Beginning

[You will need a tablet, a dollar bill, 1 dime, a penny, a calculator, and a ruler with the inch and the centimeter scales.]

EXAMPLE 1: Draw a picture of a pizza and divide it into 10 equal pieces. What fraction of the whole pizza is one piece? Label one piece 1/10.
What does the denominator of 1/10 tell us?

What does the numerator, 1, tell us?

We can write one-tenth in two different ways.
As a common fraction it is written this way: 1/10
As a decimal fraction it is written this way: 0.1

You saw how the decimal system works with whole numbers. You learned that the value of a digit depends on its place. The first place to the left of the decimal point, is the ones place. The first place to the right is the tenths place, so 0.1 is one-tenth.
The second place to the right is the hundredth place, so 0.01 is one-hundredth or 1/100.
The third place to the right is the thousandth place, so 0.001 is one-thousandth or 1/1000.

CHALLENGE 1: Darken the line around 3 pieces of the 10-piece pizza. These three pieces are what fraction of the whole pizza? Write the fraction as a common fraction and as a decimal fraction.

EXAMPLE 2: Suppose you divide a giant pizza into 100 equally-sized pieces. What decimal fraction of the whole pizza is 11 pieces?
Eleven pieces are eleven-hundredths of the whole. As a common fraction it is 11/100.
As a decimal fraction it is 0.11.

CHALLENGE 2: Write these common fractions as decimal fractions:
9/10 =
7/100 =
5/1000 =

EXAMPLE 3: Write 24.3 the "long way."
This number contains 2 tens, 4 ones, and 3 tenths.
Here is 24.3 written the "long way:" 20 + 4 + 0.3

CHALLENGE 3: Write these numbers the "long way": 5.2, 5.02, 51.26

EXAMPLE 4: Susie went to the supermarket and looked in the meat counter. She saw a package of hamburger. The label said, "Hamburger, 1.22 lb., $2.23." The weight is one and twenty-two hundredths pounds shown as a decimal number. The price is two and twenty-three hundredths dollars or two dollars twenty-three cents.

CHALLENGE 4: The label on a package hamburger says that it weighs two and fifty-five hundredths pounds. Write that amount as a decimal number. The price is five dollars seventy five cents. Write that amount as a decimal number.

EXAMPLE 5: Write 1/2 as a decimal fraction.

You must change the denominator to 10 or a multiple of 10, so you multiply it by what? If you multiply the denominator by 5 in order to get 10, what must you do to the numerator?
5/5 x 1/2 = 5/10 = 0.5 You will remember that one of the signs for division is the slash, /. We can think of a common fraction as an undone division. So 1/2 means 1 divided by 2. Do it on you calculator? Is your answer 0.5?

CHALLENGE 5: Change these common fractions to decimal fractions with your calculator and remember them:
1/4 = ?
3/4 = ?

EXAMPLE 6: Change 1/3 to a decimal fraction with the Google calculator.

In the search slot on Google simply type 1/3 and press Google Search or your Enter key. (Computers use the slash (/) for the division symbol.)

To change 1/3 to a decimal fraction, divide 1 by 3. Now isn't that interesting? The 3s keep repeating forever. Such decimals are called repeating decimals because they never stop, so you must round off at some point. Where you round off depends on the accuracy you want. For example, if your grocer makes a package of something weighing 1/3 pound, he may round down to 0.33. The most common rounding rule is that if the digit to be dropped is less than 5 you just drop it. So you round 0.333 down to 0.33. If the digit to be dropped is 5 or higher, round the number on the left up. So 0.666 is rounded up to 0.67.

Change 1/11 to a decimal fraction. What pattern is repeated?

Change 1/16 to a decimal fraction. Is the decimal number exactly the same amount as the common fraction? How many places are there to the right of the decimal point? Since the calculator can carry decimal fractions out to 14 places beyond the decimal, we know the decimal is exactly 1/16. Decimals that stop or terminate at an exact amount are called terminating decimals.
Back to the Beginning

[Go to and print out the decimal models.]

EXAMPLE: Look at decimal models (a) and (b). See if you can figure out what the rule is for placing the decimal point in the sum or remainder when adding or subtracting decimals.

The rule is to line up the decimal points one above another. This lines up the digits according to their place value.

CHALLENGE 1: Make the following additions vertically with paper and pencil:

300.3 + 3.03 = ? , 202.3 + 4.211 = ?

CHALLENGE 2: Make the following subtractions vertically with paper and pencil.

201.1 - 5.002 = ? , 6.421 - 3.02 = ?

Back to the Beginning

EXAMPLE 1: Look at decimal model (c). See if you can figure out what the rule is for placing the decimal point when multiplying numbers with decimals.

To find where to put the decimal in the product, the rule is that we count the places to the right of the decimal point in the multiplicand and add that to the number of places to the right of the decimal point in the multiplier.

If the product is dollars, we usually round to the nearest cent. For example, $5.075 is rounded to $5.08.

CHALLENGE 1: Mary bought 1.2 pounds of hamburger costing $2.03 per pound. How much did she have to pay? Round your answer to the nearest cent.

CHALLENGE 2: Do the following multiplications with paper and pencil.

10.01 x 10.1 = ? , 1.001 x 1.001 = ? , $4.55 x 2.7 pounds = ?

EXAMPLE 2: Look at decimal model (d). See if you can figure out what the rule is for placing the decimal point when dividing one decimal number by another decimal number.

The rule is to make the divisor a whole number. This was done in model (d) by moving the decimal point in the divisor one place to the right. This is the same thing as multiplying the divisor by 10. When you do something to the divisor, you must do the same thing to dividend. Therefore, you move the decimal point in the dividend one place to the right. Put the decimal point in the quotient directly above the decimal point in the dividend.

CHALLENGE 3: Mrs. Jones paid $11.25 for a 1.5 pound bag of coffee. What was the price per pound? Round to the nearest cent.

CHALLENGE 4: Make these divisions with paper and pencil rounding to the nearest cent.

78 ÷ 3.25 = ? , 45.6 ÷ 0.6 = ? , 1 ÷ 0.04 = ?

Back to the Beginning

Percent is the third kind of fraction. Let's compare percent fractions to common fractions and decimal fractions:

• Common fractions have a visible denominator that can be any numeral such as 2 in 1/2 and 10 in 1/10.
  
• Decimal fractions equal common fractions whose denominators are multiples of 10. For example, 0.5 written as a common fraction is 5/10 and 0.05 written as a common fraction is 5/100.
  
• Percent fractions have an "invisible" denominator that is always 100, so 5% written as a common fraction is 5/100 and 50% is 50/100.

Percent means "per hundred," so 5% means 5 per hundred. What does 1% mean? What does 100% mean?

If there are 100 families living in a community and 1% of them are rich, then how many families are rich? If there are 200 families and 1% of them are rich, then 2 families are rich. You can reduce 2/200 to 1/100, which is 1%.

EXAMPLE 1: Chuck has $300 in a savings account, and he receives 5% interest per year. How much interest will he receive in a year?

THINK: 5% interest means that for every $100 Chuck has in the bank he will receive $5 in interest annually. Here is the percentage formula:
Percentage (product) = Rate (factor) x Base (factor)
In this problem we know the two factors and need to compute the product.
Percentage = 5% x $300
COMPUTE: First, you need to change 5% to a decimal fraction. We simply move the decimal point two places to the left, 5% becomes 0.05. 0.05 x $300 = $15.00
CHECK: (1) He receives $5 interest on the first $100, $5 on the second, and $5 on the third, which is a total of $15. (2) Another check is to remember that we can check multiplication by division. product/factor = the other factor. So $15.00/0.05 = $300

CHALLENGE 1: Meg bought a widget for $1. The sales tax was 7%. How much did she pay including tax?

CHALLENGE 2: Gloria deposited $100 in a savings bank that pays 4% interest per year or, as they say in the trade, 4% per annum. How much interest will she earn in a year? Think about it and compute the answer in your head.

CHALLENGE 3: Capitol Clothing Store reduced the price of a $25 shirt by 40%. How much did they reduce the price?

EXAMPLE 2: Dan's Duds is having a "10% off sale." What is the sale price of a pair of trousers regularly selling for $50?

THINK: First, you could compute the discount or the amount taken "off." Second, you subtract the discount from the regular price.
COMPUTE: Reduction: 10% x $50 = $5
Sale price: $50 (old price) - $5 (reduction)= $45
CHECK: You can think of the old price as the 100% price and the new price as 90% (!00% - 10%) of the old price: 90% X $50 = 0.9 X $50 = $45.00

CHALLENGE 4: Last year there were 30 inches of rainfall in Center City. This year there was 10% less rainfall. What was the rainfall this year?

EXAMPLE 3: The teacher gave a spelling test. There were 40 words, and Tim spelled 36 of them correctly. What fraction of the words did he spell correctly, written as a percent?

THINK: In this problem you are given the percentage amount, 36, and the base, 40, and you must find the rate.

Here is our percentage formula:
Percentage (product) = Rate (factor) x Base (factor).
We need to rearrange this so that we are solving for Rate. To do this, we divide both sides of the equation by Base in order to cancel out Base on the right side, which results in this:
Rate (unknown factor) = Percentage (product) / Base (known factor)

Here is a simple way to think about it: If you know the product and one factor, you can find the unknown factor by dividing the product by the known factor:
Unknown Factor = Product ÷ Known Factor
Let's see how this works with some simple numbers:
6 (product) = 2 (factor) x (unknown factor)
Unknown factor = 6 (product) ÷ 2 (known factor) = 3

COMPUTE: On the calculator you could enter 36, division sign, 40, =, (read 0.9). Then you would change the decimal to percent, 0.9 x 100 = 90%.
But here is the simple calculator way: Enter 36 (product or percentage), division sign, 40 (known factor), % (instead of = ), read 90 or 90% (rate, which was unknown factor). How about that!

CHALLENGE 5: Jim kept $100 in his savings account. The interest he received for a year was $4. What was the interest rate paid by the bank?

EXAMPLE 4: Dan's Dandy Duds reduced the price of a jacket 20%. The sale price was $40. What was the regular price?

THINK: This is another two step problem. What is given? We are given the percent reduction (rate) and the sale price after the reduction (percentage). Let's change the percent reduction to the percent that the sale price is of the original price, 100% - 20% = 80%.
Remember the percentage formula:
Percentage (product) = Rate (factor) x Base (factor)
Now we have the percentage (product) which is $40, and the rate (known factor) which is 80%. We need to find the base or original price (unknown factor). When we know the product and one factor, what do we do to find the unknown factor?
Base (unknown factor) = Percentage (product) / Rate (known factor)
COMPUTE: Base = $40 ÷ 80% = $40 ÷ 0.8 = $50
CHECK: Review your thinking. Find the discount percentage and subtract it from the original price: $50 x 20% (0.20) = $10, $50 - $10 = $40

CHALLENGE 6: Gasoline prices were reduced 5%. The new price for regular gasoline is $1.20 per gallon. What was the original price?

EXAMPLE 5: Convert common fractions to decimals and percents.
To change a common fraction to a decimal fraction, divide the numerator by the denominator. To change a decimal fraction to a percent, multiply by 100 by simply moving the decimal point two places to the right. Remember these equivalents:

1/10 = 0.1 = 10%
1/2 = 0.5 = 50%
1-1/2 = 1.5 = 150%
1/8 = 0.125 = 12.5%

CHALLENGE 7: Convert these decimals to percents: 0.2, 0.02, 2.222

EXAMPLE 6: The price of a gallon of gasoline selling for $1.20 was increased by 10%. After a month it was reduced by 10%. What was the new price?

When you work with percents, you must be careful about the base you use.
You might think that if the price went up 10% and then went down 10% it would be the same as it was to begin with or $1.20. Right?
Wrong! The 10% increase was on the old base of $1.20. The reduction was on the new base. What is the new decreased price?

One way to do it:
1) The old base is $1.20. The increase is 10% x $1.20 = $0.12. The increased price is $1.20 + $0.12 = $1.32. 2) The new base is $1.32. The decrease is %10 x $1.32 = $0.132 (round off the 2). The decreased price is $1.32 - 13 = $1.19.
Another way to do it:
1) The old price is 100%. The increased price: 100% + 10% = 110%, so 110% x $1.20 = $1.32.
2) The new reduced price: 100% - 10% = 90%, so 90% x $1.32 = $1.19.

CHALLENGE 8: The price of bread was $1.90 per loaf. It went up 10%. A month later the price went down 10%. What was the latest price?

A ratio compares one amount to another. It may be expressed in words, 1 piece of pie to six pieces of pie, or with a ratio sign, 1:6, or as a common fraction, 1/6.

EXAMPLE 1: There are 30 children in Mr. Green's class, 18 are girls and 12 are boys. What is the ratio of the number of boys to the whole class?

The ratio of boys to the whole class is 12 to 30. We can show this as the common fraction 12/30.

CHALLENGE 1: If you cut a pie into 4 pieces and eat one piece, what is the ratio of the piece eaten to the pieces in the whole pie shown as a common fraction?

CHALLENGE 2: The Browns have 4 grandchildren; 2 are boys and 2 are girls. What is the ratio of boys to all of the grandchildren shown as a common fraction?

EXAMPLE 2: In the Brown's community, 100 babies were born during the year; 50 were boys and 50 were girls. Among the Brown's grandchildren is the ratio of boys to the total the same as the ratio in the community?
A ratio of 2 to 4 or 2/4 can be reduced to 1/2, and this is the same as 50 to 100 or 50/100 that can also be reduced to 1/2.

Now get this: Two equal ratios are a proportion. We can say that the ratio of the number of boy babies to the total Brown grandchildren is proportionate to the ratio of boy babies to the total babies in their community, 2:4::50:100.

CHALLENGE 3: Are 1/3 and 2/6 proportionate fractions?

CHALLENGE 4: Jan had a 4-inch by 6-inch picture enlarged to 8 inches by 12 inches. Is the ratio of height to width of the two pictures proportionate?

EXAMPLE 3: Let's go back to the Brown's 4 grandchildren of which 2 are boys. What percent of the children are boys?

You learned that a common fraction is ratio. We found that the ratio of boys to the total is 2/4. Now get ready for this: Percent is a proportion. A proportion is two equal ratios.
As another way to compute the percent, we can set up an equal ratio in which the numerator is the unknown percent rate and the denominator is 100.
2 is to 4 or 2/4 is equal to the ratio of the percent rate (R) to 100:
2/4 = R/100
You solve for R so put it on the left side: R/100 = 2/4
To solve for R, we must get R by itself on the left side of the equation. To do so, multiply both sides by 100 in order to cancel out the 100 on the left side. The slash (/) means the same as the division sign (÷).
R x 100 / 100 = 2 x 100 / 4
Notice that 100/100 = 1 or in other words the 100s on the left side cancel out.
Rate = 200 / 4 = 50%

CHALLENGE 5: There are 10 beans in a bottle, 5 are black. What is the ratio of black beans to the total? What is the percent of black beans? Set up a proportion and solve for the rate.