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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

withoutusing 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 ^{a}⁄_{b}). 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.

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

**(a)**^{5}⁄_{6}and^{2}⁄_{3}^{5}⁄_{6}**(b)**^{2}⁄_{3}and^{5}⁄_{7}^{5}⁄_{7}**(c)**^{13}⁄_{16}and^{5}⁄_{8}^{13}⁄_{16}**(d)**^{20}⁄_{25}and^{9}⁄_{15}^{20}⁄_{25}**(e)**^{1}⁄_{20}and^{1}⁄_{25}^{1}⁄_{20}

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

**(a)**^{5}⁄_{6}×^{2}⁄_{3}→ multiply across to get^{5 × 2}⁄_{6 × 3}=^{10}⁄_{18}→ reduce by dividing top and bottom by two to get^{5}⁄_{9}**(b)**^{3}⁄_{25}×^{10}⁄_{16}×^{5}⁄_{12}^{1}⁄_{32}**(c)**3^{1}⁄_{2}×^{1}⁄_{4}^{7}⁄_{8}**(d)**2^{5}⁄_{8}× 1^{1}⁄_{3}^{7}⁄_{2}(or write as a mixed number as 1^{1}⁄_{2})

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

**(a)**^{1}⁄_{10}÷^{1}⁄_{2}=^{1}⁄_{10}×^{2}⁄_{1}=^{2}⁄_{10}=^{1}⁄_{5}**(b)**^{7}⁄_{8}÷^{3}⁄_{8}^{7}⁄_{3}(or write as a mixed number as 2^{1}⁄_{3})**(c)**^{9}⁄_{16}÷ 2^{9}⁄_{32}**(d)**4^{1}⁄_{12}÷^{7}⁄_{8}^{14}⁄_{3}(or write as a mixed number as 4^{2}⁄_{3})

**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.

**(a)**2 ×^{5}⁄_{6}1.6666... (smaller than 2)**(b)**5 ×^{4}⁄_{5}4 (smaller than 5)**(b)**2.5 ×^{9}⁄_{10}2.25 (smaller than 2.5)**(d)**1^{1}⁄_{2}×^{2}⁄_{3}1 (smaller than 1^{1}⁄_{2})

**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.

**(a)**2 ×^{7}⁄_{6}2.3333... (bigger than 2)**(b)**5 ×^{8}⁄_{5}8 (bigger thn 5)**(c)**2.5 ×^{12}⁄_{10}3 (bigger than 2.5)**(d)**1^{1}⁄_{2}× 1^{4}⁄_{3}3.5 (bigger than 1^{1}⁄_{2})

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 answerThe 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

**(a)**Write the arithmetic statement that would tell how much sugar is needed. 1 ¾ ÷ 2**(b)**Do you expect the desired amount of sugar to be greater than or less than 1 ¾ cup? less because we are making the recipe smaller**(c)**Solve the statement (from part a). Is your answer reasonable (agrees with part b)? 1^{3}⁄_{4}÷^{2}⁄_{1}=^{7}⁄_{4}×^{1}⁄_{2}=^{7}⁄_{8}cup

**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

**(a)**How much do Margot's cats weigh total? How much vaccine is needed for the cats? 3 + 5 + 7 + 10 = 25 pounds, and then 25 × 10 = 250 milligrams**(b)**How much do Margot's dogs weigh total? How much vaccine is needed for the dogs? 12 + 14 = 28 pounds, and then 28 × 12 = 336 milligrams**(c)**How much vaccine is needed for all six pets? 250 + 336 = 586 milligrams

**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

**(a)**How much of the 36 inch length is taken up by the pictures? 5 ½ × 5 = 27 ½ inches**(b)**How much of this 36 inch length is not taken up by the pictures? 36 − 27 ½ = 8 ½ inches**(c)**If the pictures are evenly spaced as shown in the diagram below, how many inches are between each picture? 8^{1}⁄_{2}÷^{6}⁄_{1}=^{17}⁄_{2}×^{1}⁄_{6}=^{17}⁄_{12}inches

(or write as a mixed number as 1^{5}⁄_{12}inches)

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

**(a)**12 and 16 → start with 12 × 16 = 192, note the greatest common factor is 4, then do 192 ÷ 4 = 48**(b)**9 and 54 since 54*is*a multiple of 9, it is the least common multiple of the two numbers**(c)**3 and 7 start with 3 × 7 = 21, note the greatest common factor is 1, then do 21 ÷ 1 = 21**(d)**8 and 12 start with 8 × 12 = 96, note the greatest common factor is 4, then do 96 ÷ 4 = 24**(e)**24 and 60 start with 24 × 60 = 1,440, note the greatest common factor is 12, then do 1,140 ÷ 12 = 120

**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

**(a)**12 and 16 → 12 = 2 × 2 × 3 and 16 = 2 × 2 × 2 × 2 , so remove an overlap of two 2's from the latter and do 2 × 2 × 3 × 2 × 2 = 48**(b)**9 and 54 9 = 3 × 3 and 54 = 2 × 3 × 3 × 3.

All of nine's factor tree is an overlap, so remove that entire prime factorization.

What remains is the other prime factorization.

2 × 3 × 3 × 3 = 54.**(c)**3 and 7 3 and 7 are both prime.

Their prime factorizations are themselves, and have no overlap.

Multiply 3 × 7 = 21.**(d)**8 and 12 8 = 2 × 2 × 2 and 12 = 2 × 2 × 3.

The overlap is two 2's, so remove those from one prime factorization before multiplying them.

What remains is three 2's and one 3.

2 × 2 × 2 × 3 = 24.**(e)**24 and 60 24 = 2 × 2 × 2 × 3 and 60 = 2 × 2 × 3 × 5.

The overlap is two 2's and one 3, so remove those before multiplying the prime factorizations.

What remains is three 2's, one 3, and one 5.

2 × 2 × 2 × 3 × 5 = 120.

**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

**(a)**12 and 16 → 12 gets 12, 24, 36, 48, 60, 72, ... and then 16 gets 16, 32, 48 aha! a match! so the answer is 48**(b)**9 and 54 9, 18, 27, 36, 45,*54**54***(c)**3 and 7 3, 6, 9, 12, 15, 18,*21*

7, 14,*21***(d)**8 and 12 8, 16,*24*

12,*24***(e)**24 and 60 24, 48, 72, 96,*120*

60,*120*

**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

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

**(a)**^{1}⁄_{2}+^{2}⁄_{3}=^{3}⁄_{6}+^{4}⁄_{6}=^{7}⁄_{6}**(b)**^{5}⁄_{8}+^{5}⁄_{16}^{10}⁄_{16}+^{5}⁄_{16}=^{15}⁄_{16}**(c)**^{7}⁄_{10}+^{1}⁄_{12}^{42}⁄_{60}+^{5}⁄_{60}=^{47}⁄_{60}**(d)**3^{1}⁄_{4}+^{2}⁄_{3}using mixed number format 3^{3}⁄_{12}+^{8}⁄_{12}= 3^{11}⁄_{12}

or as improper fractions^{39}⁄_{12}+^{8}⁄_{12}=^{47}⁄_{12}**(e)**1^{3}⁄_{8}+ 2^{3}⁄_{4}using mixed number format 1^{3}⁄_{8}+ 2^{6}⁄_{8}= 3^{9}⁄_{8}= 4^{1}⁄_{8}

or as improper fractions^{11}⁄_{8}+^{22}⁄_{8}=^{33}⁄_{8}**(f)**^{7}⁄_{12}+^{2}⁄_{9}+^{1}⁄_{4}^{21}⁄_{36}+^{8}⁄_{36}+^{9}⁄_{36}=^{38}⁄_{36}

contuning using mixed number format^{38}⁄_{36}= 1^{2}⁄_{36}= 1^{1}⁄_{18}

or contuning as improper fractions^{38}⁄_{36}=^{19}⁄_{18}

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

**(a)**^{2}⁄_{3}−^{1}⁄_{4}=^{8}⁄_{12}−^{3}⁄_{12}=^{5}⁄_{12}**(b)**^{11}⁄_{15}−^{1}⁄_{5}^{11}⁄_{15}−^{3}⁄_{15}=^{8}⁄_{15}**(c)**1^{1}⁄_{16}−^{1}⁄_{8}^{17}⁄_{16}−^{2}⁄_{16}=^{15}⁄_{16}**(d)**2^{3}⁄_{5}−^{1}⁄_{10}2^{6}⁄_{10}−^{1}⁄_{10}= 2^{5}⁄_{10}= 2^{1}⁄_{2}

or as improper fractions^{26}⁄_{10}−^{1}⁄_{10}=^{25}⁄_{10}=^{5}⁄_{2}**(e)**10^{1}⁄_{6}− 4 6^{1}⁄_{6}**(f)**^{7}⁄_{12}−^{2}⁄_{9}−^{1}⁄_{4}^{21}⁄_{36}−^{8}⁄_{36}−^{9}⁄_{36}=^{4}⁄_{36}=^{1}⁄_{9}

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

**(a)**5^{7}⁄_{8}− 1^{3}⁄_{4}=^{47}⁄_{8}−^{7}⁄_{4}=^{47}⁄_{8}−^{14}⁄_{8}=^{33}⁄_{8}**(b)**3^{1}⁄_{2}− 2^{3}⁄_{4}^{14}⁄_{4}−^{11}⁄_{4}=^{3}⁄_{4}**(c)**6^{7}⁄_{12}− 3^{11}⁄_{12}^{79}⁄_{12}−^{47}⁄_{12}=^{32}⁄_{12}=^{8}⁄_{3}

**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

**(a)**5^{7}⁄_{8}− 1^{3}⁄_{4}= 5^{7}⁄_{8}− 1^{6}⁄_{8}→ with wholes 5 − 1 = 4 and with numerators 7 − 6 = 1 → the answer is 4^{1}⁄_{8}**(b)**3^{1}⁄_{2}− 2^{3}⁄_{4}3^{2}⁄_{4}− 2^{3}⁄_{4}= 2^{6}⁄_{4}− 2^{3}⁄_{4}=^{3}⁄_{4}**(c)**6^{7}⁄_{12}− 3^{11}⁄_{12}5^{19}⁄_{12}− 3^{11}⁄_{12}= 2^{8}⁄_{12}= 2^{1}⁄_{3}

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

**(a)**^{1}⁄_{4}→ 1 ÷ 4 = 0.25**(b)**^{1}⁄_{2}1 ÷ 2 = 0.5**(c)**^{2}⁄_{3}2 ÷ 3 = 0.6666...**(d)**^{1}⁄_{5}1 ÷ 5 = 0.2**(e)**^{4}⁄_{5}4 ÷ 5 = 0.8**(f)**^{14}⁄_{5}14 ÷ 5 = 2.8

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

**(a)**2^{1}⁄_{5}→ realize 1 ÷ 5 = 0.2 → concatenate onto the whole number to get 2.2**(b)**5^{1}⁄_{8}5.125 (the 5 sits out front, and 1 ÷ 8 = 0.125)**(c)**13^{3}⁄_{8}13.375 (the 13 sits out front, and 3 ÷ 8 = 0.375)**(d)**11^{5}⁄_{8}11.625 (the 11 sits out front, and 5 ÷ 8 = 0.625)**(e)**4^{8}⁄_{8}5 (^{8}⁄_{8}is a fifth whole)**(f)**1^{21}⁄_{8}3.625 (21 ÷ 8 = 2.625, plus the 1 that was sitting out front)

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

**(a)**^{7}⁄_{10}→ seven tenths = 0.7**(b)**^{16}⁄_{100}sixteen hundredths = 0.16**(c)**^{2}⁄_{1,000}two thousandths = 0.002**(d)**^{46}⁄_{10,000}46 ten-thousandths = 0.0046**(e)**^{138}⁄_{100}138 hundredths = 100 hundredths + 38 hundredths = 1 + 38 hundredths = 1.38**(f)**^{9,004}⁄_{1,000}9,004 thousandths = 9,000 thousandths + 4 thousandths = 9 + 4 thousandths = 9.004

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

**(a)**0.6 → six tenths →^{6}⁄_{10}→ reduce to get^{3}⁄_{5}**(b)**0.08 eight hundredths =^{8}⁄_{100}=^{2}⁄_{25}**(c)**0.125 125 thousandths =^{125}⁄_{1,000}=^{1}⁄_{8}**(d)**0.45 45 hundredths =^{45}⁄_{100}=^{9}⁄_{20}**(e)**0.4 four tenths =^{4}⁄_{10}=^{2}⁄_{5}

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

**(a)**3.6 three and six hundredths = 3^{6}⁄_{100}= 3^{3}⁄_{50}**(b)**1.075 one and 75 thousandths = 1^{75}⁄_{1,000}= 1^{3}⁄_{40}**(c)**9.0001 nine and one ten-thousandth = 9^{1}⁄_{10,000}**(d)**6.825 six and 825 thousandths = 6^{825}⁄_{1,000}= 6^{33}⁄_{40}

**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

**(a)**0.33333....^{1}⁄_{3}**(b)**0.66666....^{1}⁄_{6}**(c)**0.16666....^{2}⁄_{3}