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Section A.2 Fractions and Fraction Arithmetic

The word “fraction” comes from the Latin word fractio, which means “break into pieces”. For thousands of years, cultures from all over the world have used fractions to understand parts of a whole.
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Figure A.2.1. Alternative Video Lesson
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Subsection A.2.1 Visualizing Fractions

Parts of a Whole.

One approach to understanding fractions is to think of them as parts of a whole.
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In Figure 2, we see 1 whole divided into 7 parts. Since 3 parts are shaded, we have an illustration of the fraction 37. The denominator 7 tells us how many parts to cut up the whole; since we have 7 parts, they’re called “sevenths”. The numerator 3 tells us how many sevenths to consider.
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a rectangle that is seven times as wide as it is tall; the entire rectangle is shaded; there is a 1 in the center of the rectangle; to its right, it is labeled ’one whole’; another rectangle of the same size is aligned directly below the first rectangle; it is subdivided equally into seven squares; the first three squares are shaded; there is a 1/7 in the center of each of the first three squares; to its right, it is labeled ’three sevenths’
Figure A.2.2. Representing 37 as parts of a whole.

Checkpoint A.2.3. A Fraction as Parts of a Whole.

To visualize the fraction 1435, you might cut a rectangle into equal parts, and then count up of them.
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Explanation.
You could cut a rectangle into 35 equal pieces, and then 14 of them would represent 1435.
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We can also locate fractions on number lines. When ticks are equally spread apart, as in Figure 4, each tick represents a fraction.
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a number line with a curved arrow emanating from 0 and landing to the right at 1/7, then again from 1/7 to 2/7, and again from 2/7 to 3/7
Figure A.2.4. Representing 37 on a number line.

Checkpoint A.2.5. A Fraction on a Number Line.

In the given number line, what fraction is marked?
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a number line with -1, 0, and 1 marked; there are evenly spaced ticks, with eight ticks between -1 and 0, eight between 0 and 1, and so on; there is a dot marked at the fifth tick to the right from 0
Explanation.
There are 8 subdivisions between 0 and 1, and the mark is at the fifth subdivision. So the mark is 58 of the way from 0 to 1 and therefore represents the fraction 58.
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Division.

Fractions can also be understood through division.
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For example, we can view the fraction 37 as 3 divided into 7 equal parts, as in Figure 6. Just one of those parts represents 37.
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a number line with a ruler superimposed over the segment from 0 to 3; the ruler is subdivided into seven pieces; an arrow points down from the first ruler tick to a point on the number line that is marked 3/7
Figure A.2.6. Representing 37 on a number line.

Checkpoint A.2.7. Seeing a Fraction as Division Arithmetic.

The fraction 2140 can be thought of as dividing the whole number into equal-sized parts.
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Explanation.
Since 2140 means the same as 21÷40, it can be thought of as dividing 21 into 40 equal parts.
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Subsection A.2.2 Equivalent Fractions

It’s common to have two fractions that represent the same amount. Consider 25 and 615 represented in various ways in Figures 8–Figure 10.
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one rectangle that is five times as wide as it is tall; the entire rectangle is shaded; there is a 1 in the center of the rectangle; a second rectangle that is also five times as wide as it is tall; it is subdivided equally into five squares; the first two squares are shaded; there is a 1/5 in the center of each of the first two squares; a third rectangle that is also five times as wide as it is tall; it is subdivided equally into fifteen adjacent rectangles; the first six of these smaller rectangles are shaded; there is a 1/15 in the center of each of the first six smaller rectangles
Figure A.2.8. 25 and 615 as equal parts of a whole
a number line with a curved arrow emanating from 0 and landing to the right at 1/5, then again from 1/5 to 2/5; there is a number line with a curved arrow emanating from 0 and landing to the right at 1/15, then again from 1/15 to 2/15, then again from 2/15 to 3/15, then again from 3/15 to 4/15, then again from 4/15 to 5/15, and then again from 5/15 to 6/15
Figure A.2.9. 25 and 615 as equal on a number line
a number line with a ruler superimposed over the segment from 0 to 2; the ruler is subdivided into five pieces; an arrow points down from the first ruler tick to a point on the number line that is marked 2/5; there is a second number line with a ruler superimposed over the segment from 0 to 6; the ruler is subdivided into fifteen pieces; an arrow points down from the first ruler tick to a point on the number line that is marked 6/15
Figure A.2.10. 25 and 615 as equal quotients
Those two fractions, 25 and 615 are equal, as those figures demonstrate. In addition, both fractions are equal to 0.4 as a decimal. If we must work with this number, the fraction that uses smaller numbers, 25, is preferable. Working with smaller numbers decreases the likelihood of making a human arithmetic error and it also increases the chances that you might make useful observations about the nature of that number.
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So if you are handed a fraction like 615, it is important to try to reduce it to “lowest terms”. The most important skill you can have to help you do this is to know the multiplication table well. If you know it well, you know that 6=2⋅3 and 15=3⋅5, so you can break down the numerator and denominator that way. Both the numerator and denominator are divisible by 3, so they can be “factored out” and then as factors, cancel out.
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615=2⋅33⋅5=2⋅33⋅5=2⋅11⋅5=25
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Checkpoint A.2.11. Reducing Fractions.

Reduce these fractions into lowest terms.
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(a)

1442=
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Explanation.
With 1442, we have 2⋅72⋅3⋅7, which reduces to 13.

(b)

830=
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Explanation.
With 830, we have 2⋅2⋅22⋅3⋅5, which reduces to 415.

(c)

7090=
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Explanation.
With 7090, we have 7⋅109⋅10, which reduces to 79.
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Sometimes it is useful to do the opposite of reducing a fraction, and build up the fraction to use larger numbers.
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Checkpoint A.2.12. Building Up a Fraction.

Sayid scored 2125 on a recent exam. Build up this fraction so that the denominator is 100, so that Sayid can understand what percent score he earned.
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Explanation.
To change the denominator from 25 to 100, it needs to be multiplied by 4. So we calculate
2125=21⋅425⋅4=84100
So the fraction 2125 is equivalent to 84100. (This means Sayid scored an 84%.)
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Subsection A.2.3 Multiplying with Fractions

Example A.2.13.

Suppose a recipe calls for 23 cup of milk, but we’d like to quadruple the recipe (make it four times as big). We’ll need four times as much milk, and one way to measure this out is to fill a measuring cup to 23 full, four times:
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four measuring cups, each filled two-thirds with liquid
When you count up the shaded thirds, there are eight of them. So multiplying 23 by the whole number 4, the result is 83. Mathematically:
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4⋅23=4⋅23=83
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Example A.2.15.

We could also use multiplication to decrease amounts. Suppose we needed to cut the recipe down to just one fifth. Instead of four of the 23 cup milk, we need one fifth of the 23 cup milk. So instead of multiplying by 4, we multiply by 15. But how much is 15 of 23 cup?
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If we cut the measuring cup into five equal vertical strips along with the three equal horizontal strips, then in total there are 3⋅5=15 subdivisions of the cup. Two of those sections represent 15 of the 23 cup.
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a measuring cup divided evenly into three horizontal sections, but also evenly into five vertical sections; in total, there are fifteen sections; the two that correspond to the lower two-thirds and leftmost one-fifth are shaded
In the end, we have 215 of a cup. The denominator 15 came from multiplying 5 and 3, the denominators of the fractions we had to multiply. The numerator 2 came from multiplying 1 and 2, the numerators of the fractions we had to multiply.
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15⋅23=1⋅25⋅3=215
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Checkpoint A.2.17. Fraction Multiplication.

Simplify these fraction products.
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(a)

13⋅107=
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Explanation.
Multiplying numerators gives 10, and multiplying denominators gives 21. The answer is 1021.

(b)

123⋅153=
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Explanation.
Before we multiply fractions, note that 123 reduces to 4, and 153 reduces to 5. So we just have 4⋅5=20.

(c)

−145⋅23=
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Explanation.
Multiplying numerators gives 28, and multiplying denominators gives 15. The result should be negative, so the answer is −2815.

(d)

7027⋅12−20=
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Explanation.
Before we multiply fractions, note that 12−20 reduces to −35. So we have 7027⋅−35. Both the numerator of the first fraction and denominator of the second fraction are divisible by 5, so it helps to reduce both fractions accordingly and get 1427⋅−31. Both the denominator of the first fraction and numerator of the second fraction are divisible by 3, so it helps to reduce both fractions accordingly and get 149⋅−11. Now we are just multiplying 149 by −1, so the result is −149.
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Subsection A.2.4 Division with Fractions

How does division with fractions work? Are we able to compute/simplify each of these examples?
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  1. 3÷27
  2. 1819÷5
  3. 143÷89
  4.  25 52
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We know that when we divide something by 2, this is the same as multiplying it by 12. Conversely, dividing a number or expression by 12 is the same as multiplying by 21, or just 2. The more general property is that when we divide a number or expression by ab, this is equivalent to multiplying by the reciprocal ba.
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Example A.2.19.

With our examples from the beginning of this subsection:
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  1. 3÷27=3⋅72=31⋅72=212 
  2. 1819÷5=1819÷51=1819⋅15=1895 
  3. 143÷89=143⋅98=141⋅38=71⋅34=214
  4.  25 52=25÷52=25⋅25=425 
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Checkpoint A.2.20. Fraction Division.

Simplify these fraction division expressions.
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(a)

13÷107=
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Explanation.
13÷107=13⋅710=730 

(b)

125÷5=
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Explanation.
125÷5=125⋅15=1225 

(c)

−14÷32=
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Explanation.
−14÷32=−14⋅23=−141⋅23=−283

(d)

709÷11−20=
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Explanation.
709÷11−20=−709⋅2011=−140099 
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Subsection A.2.5 Adding and Subtracting Fractions

With whole numbers and integers, operations of addition and subtraction are relatively straightforward. The situation is almost as straightforward with fractions if the two fractions have the same denominator. Consider
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72+32=7 halves+3 halves
In the same way that 7 tacos and 3 tacos make 10 tacos, we have:
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7 halves+3 halves=10 halves72+32=102=5
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Checkpoint A.2.22. Fraction Addition and Subtraction.

Add or subtract these fractions.
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(a)

13+103=
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Explanation.
Since the denominators are both 3, we can add the numerators: 1+10=11. The answer is 113.

(b)

136−56=
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Explanation.
Since the denominators are both 6, we can subtract the numerators: 13−5=8. The answer is 86, but that reduces to 43.
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Whenever we’d like to combine fractional amounts that don’t represent the same number of parts of a whole (that is, when the denominators are different), finding sums and differences is more complicated.
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Example A.2.23. Quarters and Dimes.

Find the sum 34+210. Does this seem intimidating? Consider this:
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  • 14 of a dollar is a quarter, and so 34 of a dollar is 75 cents.
  • 110 of a dollar is a dime, and so 210 of a dollar is 20 cents.
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So if you know what to look for, the expression 34+210 is like adding 75 cents and 20 cents, which gives you 95 cents. As a fraction of one dollar, that is 95100. So we can report
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34+210=95100.
(Although we should probably reduce that last fraction to 1920.)
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This example was not something you can apply to other fraction addition situations, because the denominators here worked especially well with money amounts. But there is something we can learn here. The fraction 34 was equivalent to 75100, and the other fraction 210 was equivalent to 20100. These equivalent fractions have the same denominator and are therefore “easy” to add. What we saw happen was:
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34+210=75100+20100=95100
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This realization gives us a strategy for adding (or subtracting) fractions.
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Example A.2.25.

Let’s add 23+25. The denominators are 3 and 5, so the number 15 would be a good common denominator.
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23+25=2⋅53⋅5+2⋅35⋅3=1015+615=1615
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Checkpoint A.2.26. Using Some Flour.

A chef had 23 cups of flour and needed to use 18 cup to thicken a sauce. How much flour is left?
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Explanation.
We need to compute 23−18. The denominators are 3 and 8. One common denominator is 24, so we move to rewrite each fraction using 24 as the denominator:
23−18=2⋅83⋅8−1⋅38⋅3=1624−324=1324
The numerical result is 1324, but a pure number does not answer this question. The amount of flour remaining is 1324 cups.
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Subsection A.2.6 Mixed Numbers and Improper Fractions

A simple recipe for bread contains only a few ingredients:
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11/2 tablespoons yeast
11/2 tablespoons kosher salt
61/2 cups unbleached, all-purpose flour (more for dusting)
Each ingredient is listed as a mixed number that quickly communicates how many whole amounts and how many parts are needed. It’s useful for quickly communicating a practical amount of something you are cooking with, measuring on a ruler, purchasing at the grocery store, etc. But it causes trouble in an algebra class. The number 11/2 means “one and one half”. So really,
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112=1+12
The trouble is that with 11/2, you have two numbers written right next to each other. Normally with two math expressions written right next to each other, they should be multiplied, not added. But with a mixed number, they should be added.
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Fortunately we just reviewed how to add fractions. If we need to do any arithmetic with a mixed number like 11/2, we can treat it as 1+12 and simplify to get a “nice” fraction instead: 32. A fraction like 32 is called an improper fraction because it’s actually larger than 1. And a “proper” fraction would be something small that is only part of a whole instead of more than a whole.
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112=1+12=11+12=22+12=32
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Exercises A.2.7 Exercises

Review and Warmup.

1.

Which letter is −214 on the number line?
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2.

Which letter is 274m on the number line?
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  • ?
  • A
  • B
  • C
  • D
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3.

The dot in the graph can be represented by what fraction?
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4.

The dot in the graph can be represented by what fraction?
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5.

The dot in the graph can be represented by what fraction?
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6.

The dot in the graph can be represented by what fraction?
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Reducing Fractions.

7.

Reduce the fraction 212.
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8.

Reduce the fraction 930.
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9.

Reduce the fraction 63147.
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10.

Reduce the fraction 2227.
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11.

Reduce the fraction 385210.
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12.

Reduce the fraction 546.
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Building Fractions.

13.

Find an equivalent fraction to 19 with denominator 27.
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14.

Find an equivalent fraction to 12 with denominator 10.
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15.

Find an equivalent fraction to 25 with denominator 15.
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16.

Find an equivalent fraction to 17 with denominator 28.
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Multiplying/Dividing Fractions.

17.

Multiply: 37⋅18
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18.

Multiply: 37⋅310
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19.

Multiply: 1011⋅52
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20.

Multiply: 127⋅116
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23.

Multiply: −411⋅138
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24.

Multiply: −52⋅310
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25.

Multiply: 30⋅(−310)
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26.

Multiply: 35⋅(−47)
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27.

Multiply: 225⋅214⋅59
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28.

Multiply: 1425⋅349⋅59
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29.

Multiply: 35⋅79⋅10
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30.

Multiply: 27⋅34⋅35
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33.

Divide: 58÷(−320)
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34.

Divide: 16÷(−710)
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35.

Divide: −157÷(−10)
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36.

Divide: −152÷(−10)
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39.

Multiply: 2112⋅11335
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40.

Multiply: 279⋅1225
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Adding/Subtracting Fractions.

65.

Subtract: 1918−1718
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66.

Subtract: 2140−940
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67.

Subtract: 57−542
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68.

Subtract: 38−1732
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69.

Subtract: 3245−19
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70.

Subtract: 1114−27
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71.

Subtract: −110−16
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72.

Subtract: −16−710
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73.

Subtract: −310−(−56)
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74.

Subtract: −56−(−310)
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75.

Subtract: 4−136
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76.

Subtract: 5−173
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Applications.

77.

Anthony walked 14 of a mile in the morning, and then walked 15 of a mile in the afternoon. How far did Anthony walk altogether?
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Anthony walked a total of of a mile.
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78.

Joseph walked 15 of a mile in the morning, and then walked 18 of a mile in the afternoon. How far did Joseph walk altogether?
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Joseph walked a total of of a mile.
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79.

Tammy and Hannah are sharing a pizza. Tammy ate 16 of the pizza, and Hannah ate 110 of the pizza. How much of the pizza was eaten in total?
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They ate of the pizza.
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80.

A trail’s total length is 3263 of a mile. It has two legs. The first leg is 29 of a mile long. How long is the second leg?
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The second leg is of a mile in length.
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81.

A trail’s total length is 2363 of a mile. It has two legs. The first leg is 29 of a mile long. How long is the second leg?
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The second leg is of a mile in length.
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82.

Jon is participating in a running event. In the first hour, he completed 16 of the total distance. After another hour, in total he had completed 724 of the total distance.
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What fraction of the total distance did Jon complete during the second hour?
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Jon completed of the distance during the second hour.
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83.

The pie chart represents a school’s student population.
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Together, white and black students make up of the school’s population.
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84.

Each page of a book is 814 inches in height, and consists of a header (a top margin), a footer (a bottom margin), and the middle part (the body). The header is 56 of an inch thick and the middle part is 656 inches from top to bottom.
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What is the thickness of the footer?
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The footer is of an inch thick.
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85.

Donna and Laney are sharing a pizza. Donna ate 15 of the pizza, and Laney ate 16 of the pizza. How much more pizza did Donna eat than Laney?
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Donna ate more of the pizza than Laney ate.
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86.

Douglas and Katherine are sharing a pizza. Douglas ate 15 of the pizza, and Katherine ate 110 of the pizza. How much more pizza did Douglas eat than Katherine?
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Douglas ate more of the pizza than Katherine ate.
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87.

A school had a fund-raising event. The revenue came from three resources: ticket sales, auction sales, and donations. Ticket sales account for 13 of the total revenue; auction sales account for 35 of the total revenue. What fraction of the revenue came from donations?
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of the revenue came from donations.
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88.

A few years back, a car was purchased for $12,000. Today it is worth 14 of its original value. What is the car’s current value?
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The car’s current value is .
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89.

A few years back, a car was purchased for $17,600. Today it is worth 14 of its original value. What is the car’s current value?
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The car’s current value is .
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90.

The pie chart represents a school’s student population.
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more of the school is white students than black students.
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91.

A town has 250 residents in total, of which 45 are Latino Americans. How many Latino Americans reside in this town?
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There are Latino Americans residing in this town.
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92.

A company received a grant, and decided to spend 1112 of this grant in research and development next year. Out of the money set aside for research and development, 1011 will be used to buy new equipment. What fraction of the grant will be used to buy new equipment?
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of the grant will be used to buy new equipment.
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93.

A food bank just received 38 kilograms of emergency food. Each family in need is to receive 25 kilograms of food. How many families can be served with the 38 kilograms of food?
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families can be served with the 38 kilograms of food.
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94.

A construction team maintains a 72-mile-long sewage pipe. Each day, the team can cover 910 of a mile. How many days will it take the team to complete the maintenance of the entire sewage pipe?
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It will take the team days to complete maintaining the entire sewage pipe.
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95.

A child is stacking up tiles. Each tile’s height is 23 of a centimeter. How many layers of tiles are needed to reach 12 centimeters in total height?
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To reach the total height of 12 centimeters, layers of tiles are needed.
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96.

A restaurant made 150 cups of pudding for a festival.
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Customers at the festival will be served 16 of a cup of pudding per serving. How many customers can the restaurant serve at the festival with the 150 cups of pudding?
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The restaurant can serve customers at the festival with the 150 cups of pudding.
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97.

A 2×4 piece of lumber in your garage is 4818 inches long. A second 2×4 is 69132 inches long. If you lay them end to end, what will the total length be?
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The total length will be inches.
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98.

A 2×4 piece of lumber in your garage is 6518 inches long. A second 2×4 is 58916 inches long. If you lay them end to end, what will the total length be?
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The total length will be inches.
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99.

Each page of a book consists of a header, a footer and the middle part. The header is 16 inches in height; the footer is 1112 inches in height; and the middle part is 616 inches in height.
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What is the total height of each page in this book? Use mixed number in your answer if needed.
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Each page in this book is inches in height.
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100.

To pave the road on Ellis Street, the crew used 134 tons of cement on the first day, and used 249 tons on the second day. How many tons of cement were used in all?
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tons of cement were used in all.
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101.

When driving on a high way, noticed a sign saying exit to Johnstown is 134 miles away, while exit to Jerrystown is 414 miles away. How far is Johnstown from Jerrystown?
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Johnstown and Jerrystown are miles apart.
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102.

A cake recipe needs 118 cups of flour. Using this recipe, to bake 6 cakes, how many cups of flour are needed?
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To bake 6 cakes, cups of flour are needed.
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Sketching Fractions.

105.

Sketch a picture of the product 35⋅12, using a number line or rectangles.
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106.

Sketch a picture of the sum 23+18, using a number line or rectangles.
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Challenge.

107.

Given that a≠0, simplify 9a+8a.
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108.

Given that a≠0, simplify 1a+52a.
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109.

Given that a≠0, simplify 2a−15a.
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You have attempted 1 of 2 activities on this page.
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