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Math Lecture Notes
Fraction Review Problems

The first few days of our Math 20 class are for reviewing foundational topics about arithmetic and fractions.

The homework problems below hope to provide an interesting, thought-provoking, and useful review of fractions. (The arithmetic review problems are here.)

As always, this is merely a "best of their kind" collection of problems. If your math background is strong you might feel there are too many. If your math background is rusty or weak you might need to do additional problems from the textbook to get enough review and practice.

Try to work these problems without using a calculator. Calculators will not be allowed on the midterm that covers these review topics.

Use the floating buttons in the bottom right corner of the screen to show or hide the answers. Before you look at the answers you should have tried to work the problems "forwards". Do not examine the answers first and then try to work "backwards". For these foundational review topics it is better to wait and ask questions during class.

When you have done enough review, try the fraction homework.

Note: Due to the limitations of internet browsers, the fractions on this page are written diagonally (like ab). Do not do this! You are writing on paper, so write your fractions vertically, with the numerator directly above the denominator. The habit of writing fractions verticaly makes fraction canceling and arithmetic more visually intuitive. Also, it will later make working with ratios and equations much easier.

Desired Denominators link to here link to index

Here is a quick task to try before doing arithmetic with fractions.

The online notes for this topic are here.

1. Which fraction from each list is greatest? (Hint: rewrite them with common denominators.) §2.5

Fraction Multiplication and Division link to here link to index

We start with the easiest fraction arithmetic. No common denominators required!

Fraction multiplication is the foundation of the Unit Analysis technique we will use to do complicated measurement unit conversions.

The online notes for this topic are here.

1. Multiply. Remember to reduce your answers if you need to. (Hint: use canceling to minimize reducing.) §2.6

2. Divide. Remember to reduce your answers if you need to. (Hint: use canceling to minimize reducing.) §2.7

3. Multiply using a calculator to get a decimal answer. When we multiply by a proper fraction, does it make the starting number bigger or smaller? §3.6 Multiplying by a proper fraction makes the starting number smaller.

4. Multiply using a calculator to get a decimal answer. When we multiply by an improper fraction, does it make the starting number bigger or smaller? §3.6 Multiplying by an improper fraction makes the starting number bigger.

The Six Step Problem Solving Method link to here link to index

Now that we have reviewed fraction multiplication and division, we can attempt more interesting word problems.

Recall the six-step problem solving process.

1. Determine what you are looking for
2. Draw pictures
3. Name things
4. Make equations
5. Solve the equations
6. Check your answer

The following word problems hold your hand a bit. Think carefully about how the six-step problem solving process applies to them, to help you understand the six steps later when there is no hand-holding.

The online notes for this topic are here.

1. A recipe calls for 1 ¾ cups of sugar. Suppose you want to cut the recipe in half. §1.8, §2.7

2. The Deltallution vaccine keeps fleas off pets. Cats need a 10 milligram dose per pound. Dogs need a 12 milligram dose per pound. Margot owns four cats (weighing 3, 5, 7, and 10 pounds) and two dogs (weighing 12 and 14 pounds). How many milligrams of vaccine would be needed to vaccinate all six pets? §1.8, §2.7

3. Five small pictures, each measuring 5 ½ inches, are being framed by cutting five squares from a 36 inch long matte. §1.8, §2.6, §2.7, §3.2, §3.3

matte diagram

Least Common Multiples link to here link to index

Before adding or subtracting fractions we need to find a common denominator. The most friendly number to use is the least common multiple of the old denominators.

There are no online notes for this topic.

1. Use the brute force method to find the least common multiple of each pair of numbers. Start by multiplying the two numbers, then divide by their greatest common factor. §2.1, §3.1

2. Use the factor tree method to find the least common multiple of each pair of numbers. Cross out overlaps from the prime factorizations, then multiply what is left. §2.1, §3.1

3. Use the list of multiples method to find the least common multiple of each pair of numbers. List multiples of each number until you see a match. §2.1, §3.1

4. When two numbers are relatively prime, what is their least common multiple? §1.5 their product

5. When one number is a factor of another number, what is their least common multiple? §1.5 the larger number

Fraction Addition and Subtraction link to here link to index

We review fraction addition and subtraction in preparation for Math 60.

None of the Math 20 topics reinforce this review topic. Fraction addition and subtraction never occurs in Math 20 topics after this review time. So study this topics and keep it accessible in the back of your head, but realize you will need to plan time to review again as you approach Math 60.

It is okay to leave improper fraction answers in that format. If your future includes mathematics or computer programming you will prefer improper fractions. If your future includes mathematics or computer programming you will prefer mixed numbers. At this point in your math career feel free to use whichever you like best.

The online notes for this topic are here.

1. Add. Remember to reduce your answers if you need to. §3.2

2. Subtract. Remember to reduce your answers if you need to. §3.3

Subtracting Mixed Numbers link to here link to index

The idea that you can think of a mixed number in a place value manner will later on help change a mixed number into percent format.

The online notes for this topic are here.

1. Subtract by changing the mixed numbers into improper fractions. Remember to reduce your answers if you need to. §3.5

2. Subtract by treating the whole numbers and fractions like place value columns, and borrowing when necessary. Remember to reduce your answers if you need to. §3.5

Changing Fractions to Decimals link to here link to index

Later we will extend this topic. We will change fractions first into decimal format and then into percent format.

The online notes for this topic are here.

1. Change these fractions into decimals. (Hint: You can always use long division.) §4.5

2. Change these mixed numbers into decimals. (Hint: Do not do too much with the whole number.) §4.5

3. Change these fractions into decimals. (Hint: There is a shortcut. But you can always use long division.) §4.5

Changing Decimals to Fractions link to here link to index

Later we will extend this topic. We will change percentages first into decimal format and then into fractions.

The online notes for this topic are here.

1. Change these decimals into fractions. Remember to reduce your answers if you need to. §4.1

2. Change these decimals into fractions. Remember to reduce your answers if you need to. §4.1

3. Change these decimals into fractions. (These are the three you will often use this term for which memorization helps.) Remember to reduce your answers if you need to. §4.1