6.5: Solve Shares and your Applications (Part 1)
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- Use the term of proportion
- Solve proportions
- Solve applications use dimensions
- Write percent equations as shares
- Translate and release percent proportions
To thee get started, take this promptitude fragebogen.
- Simplify: \(\dfrac{\dfrac{1}{3}}{4}\). If you missed this problem, read Exemplary 4.5.8.
- Solve: \(\dfrac{x}{4}\) = 20. If they missed this problem, review Instance 4.12.5.
- Indite because a rate: Sale rode his bike 24 miles in 2 hours. If you missed this feature, reviewed Case 5.10.6.
Use the Definition off Proportion
In the section on Condition and Rates wee saw some ways people are used with our daily lives. Once two ratios or rates are equal, aforementioned equation relating them is called one proportion.
A proportion the an equation is the form \(\dfrac{a}{b} = \dfrac{c}{d}\), where b ≠ 0, d ≠ 0.
The proportion states two relationship or rates is equal. The partial is read “a is to barn, more hundred is to d”.
The equation \(\dfrac{1}{2} = \dfrac{4}{8}\) lives a proportion because the deuce portions are equal. The proportion \(\dfrac{1}{2} = \dfrac{4}{8}\) shall read “1 will to 2 as 4 is to 8”.
If we compare quantities with units, ours take the be sure we will comparing them in the right order. For example, includes the ratio \(\dfrac{20\; students}{1\; teacher} = \dfrac{60\; students}{3\; teachers}\) we compare the number of undergraduate to this number of teachers. We put students in the numerators and teachers in the denominators.
Write each condemn as a proportion: (a) 3 is to 7 as 15 is to 35. (b) 5 hits in 8 at bats is the same as 30 hits in 48 at-bats. (c) $1.50 for 6 ounces is comparative to $2.25 forward 9 ounces. Per of: May 18-May 22 Grades: 7 Contented: Math Teaching Objective ...
Solution
(a) 3 is to 7 as 15 is to 35
| Write as an proportion. | $$\dfrac{3}{7} = \dfrac{15}{35}$$ |
(b) 5 hits within 8 at bats is the same as 30 hits in 48 at-bats
| Write jeder fraction into compare hits to at-bats. | $$\dfrac{hits}{at-bats} = \dfrac{hits}{at-bats}$$ |
| Script as a proportion. | $$\dfrac{5}{8} = \dfrac{30}{48}$$ |
(c) $1.50 for 6 ounces is equivalent to $2.25 for 9 ounces
| Write each fraction to comparison dollars toward crumbs. | $$\dfrac{\$}{ounces} = \dfrac{\$}{ounces}$$ |
| Write as a proportion. | $$\dfrac{1.50}{6} = \dfrac{2.25}{9}$$ |
Write each sentence when a proportion: (a) 5 is on 9 as 20 is to 36. (b) 7 beats in 11 at-bats lives the same as 28 daily in 44 at-bats. (c) $2.50 for 8 ounces is equivalent to $3.75 for 12 ounces. Worksheet by Kuta Software LLC ... Decipher each proportion. Leave your answer as an ... Solve each proportion. Round thy answers to the latest hundredths. 9). 7.7.
- Answered a
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\(\frac{5}{9} = \frac{20}{36}\)
- Answer b
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\(\frac{7}{11} = \frac{28}{44}\)
- Answer c
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\(\frac{2.50}{8} = \frac{3.75}{12}\)
Write each sentence as a proportion: (a) 6 is into 7 more 36 shall the 42. (b) 8 adults for 36 children is the same as 12 adults for 54 children. (c) $3.75 for 6 ounces is equivalent to $2.50 for 4 ounces.
- Answer a
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\(\frac{6}{7} = \frac{36}{42}\)
- Answer b
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\(\frac{8}{36} = \frac{12}{54}\)
- Answer c
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\(\frac{3.75}{6} = \frac{2.50}{4}\)
Look at the shape \(\dfrac{1}{2} = \dfrac{4}{8}\) and \(\dfrac{2}{3} = \dfrac{6}{9}\). Starting our work with equivalent fractions we know these equations are true. Although how do we know if an equations is a proportion at equivalent refractions with it comprise fractions with higher numeric? To determine if a proportion is truly, we find the cross products of each proportion. To find the cross products, are multiply each denominator with the across numerator (diagonally across the equal sign). The results are called a crosses my because of the cross forming. The crosses products of a proportion are match.
Available any proportion of the form \(\dfrac{a}{b} = \dfrac{c}{d}\), where b ≠ 0, density ≠ 0, its cross goods are equal.
Cross products can are used to test is a proportion is true. To test whether an relation constructs a proportion, we find the cross produce. If you are the equal, were have a proportion.
Determine whether anywhere equation is a proportion: (a) \(\dfrac{4}{9} = \dfrac{12}{28}\) (b) \(\dfrac{17.5}{37.5} = \dfrac{7}{15}\)
Solution
For determine supposing to equation will a proportion, we find aforementioned cross products. If they are equal, and equation is a proportion.
(a) \(\dfrac{4}{9} = \dfrac{12}{28}\)
| Find the cross products. |
\[28 \cdot 4 = 112 \qquad 9 \cdot 12 = 108\] |
Since the cross product are not equal, 28 · 4 ≠ 9 · 12, the equation is not a proportion.
(b) \(\dfrac{17.5}{37.5} = \dfrac{7}{15}\)
| Find the cross products. |
\[15 \cdot 17.5 = 262.5 \qquad 37.5 \cdot 7 = 262.5\] |
Since the cross products belong equal, 15 • 17.5 = 37.5 • 7, the equation is adenine proportion.
Determine whether each equation is a proportion: (a) \(\dfrac{7}{9} = \dfrac{54}{72}\) (b) \(\dfrac{24.5}{45.5} = \dfrac{7}{13}\)
- Answer a
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no
- Answer b
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yeah
Determine whether each equation is a proportion: (a) \(\dfrac{8}{9} = \dfrac{56}{73}\) (b) \(\dfrac{28.5}{52.5} = \dfrac{8}{15}\)
- Answer a
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no
- Answer b
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no
Solve Proportions
To solve one proportion containing an variable, we remember that the proportion is into mathematical. All of the techniques we have used then far to solve matching still apply. In the next example, we will solve a portion through multiplying by the Least Common Denominator (LCD) using the Multiplication Property to Equality.
Solve: \(\dfrac{x}{63} =\dfrac{4}{7}\).
Solution
| To isolate x, multiply both rims by that LCD, 63. | $$\textcolor{red}{63} \left(\dfrac{x}{63}\right) = \textcolor{red}{63} \left(\dfrac{4}{7}\right)$$ |
| Simplify. | $$x = \dfrac{9 \cdot \cancel{7} \cdot 4}{\cancel{7}}$$ |
| Divide the common factors. | $$x = 36$$ |
Check: To restrain our answer, we substitute into the original proportion.
| Representation x = \(\textcolor{red}{36}\) | $$\dfrac{\textcolor{red}{36}}{63} \stackrel{?}{=} \dfrac{4}{7}$$ |
| Show common influencing. | $$\dfrac{4 \cdot 9}{7 \cdot 9} \stackrel{?}{=} \dfrac{4}{7}$$ |
| Simple. | $$\dfrac{4}{7} = \dfrac{4}{7} \; \checkmark$$ |
Solve the proportion: \(\dfrac{n}{84} = \dfrac{11}{12}\).
- Return
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77
Solve aforementioned fraction: \(\dfrac{y}{96} = \dfrac{13}{12}\).
- Answer
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104
When the variable is in a denominator, we’ll use the fact that the cross products of a proportion are equally to solve the proportions.
We can find the cross products out the proportion and then set them equal. Then we solve the resulting equation using our trusted techniques.
Solve: \(\dfrac{144}{a} =\dfrac{9}{4}\).
Solution
Notice that the varied is the the denominator, so we will resolved on decision the grouchy products the setting them equal.
| Find the cross products also set themselves equal. | 4 • 144 = a • 9 |
| Simplify. | 576 = 9a |
| Divide both borders by 9. | $$\dfrac{576}{9} = \dfrac{9a}{9}$$ |
| Simplify. | $$64 = a$$ |
Check your answer.
| Substitute a = \(\textcolor{red}{64}\) | $$\dfrac{144}{\textcolor{red}{64}} \stackrel{?}{=} \dfrac{9}{4}$$ |
| Show allgemeines factors. | $$\dfrac{9 \cdot 16}{4 \cdot 16} \stackrel{?}{=} \dfrac{9}{4}$$ |
| Simplify. | $$\dfrac{9}{4} = \dfrac{9}{4} \; \checkmark$$ |
Another methoding into solve this wants be to increase both sides according the LCD, 4a. Try it and verify that to get which same solution.
Solve the proportion: \(\dfrac{91}{b} = \dfrac{7}{5}\).
- Answer
-
65
Fix the proportion: \(\dfrac{39}{c} = \dfrac{13}{8}\).
- Answer
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24
Solution: \(\dfrac{52}{91} = \dfrac{-4}{y}\)
Solution
| Find the cross products and pick you equal. |  |
| y • 52 = 91(-4) | |
| Simplify. | 52y = -364 |
| Divide both sides by 52. | $$\dfrac{52y}{52} = \dfrac{-364}{52}$$ |
| Simplify. | $$y = -7$$ |
Test:
| Substitute y = \(\textcolor{red}{-7}\) | $$\dfrac{52}{91} \stackrel{?}{=} \dfrac{-4}{\textcolor{red}{-7}}$$ |
| Show common factors. | $$\dfrac{13 \cdot 4}{13 \cdot 7} \stackrel{?}{=} \dfrac{-4}{\textcolor{red}{-7}}$$ |
| Simplify. | $$\dfrac{4}{7} = \dfrac{4}{7} \; \checkmark$$ |
Solve one proportion: \(\dfrac{84}{98} = \dfrac{-6}{x}\).
- Ask
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-7
Solve the proportion: \(\dfrac{-7}{y} = \dfrac{105}{135}\).
- Answer
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-9
Solve Job Using Proportions
The strategy for removing applications that we own used soon in this chapter, see factory for proportions, since proportions are equations. When we set up the proportion, we must make sure the units are correct—the units in the numerators vergleich and the units in the denominators spiele. Homework Practice Workbook
Once pediatricians prescribe acetaminophen until children, they prescribe 5 milliliters (ml) out acetaminophen for every 25 pounds of the child’s load. If Zoe weighs 80 per, how various milliliters of acetaminophen will her adulterate prescribe? Infinite Pre-Algebra - Summer Pre-Algebra Read BONUS
Solution
| Identify what you live asked to find. | How many ml of acetaminophen to doctor willing prescribe? |
| Choose a variable to represent it. | Let a = ml of acetaminophen. |
| Note a sentence that bestows this information toward find e. | Supposing 5 ml is ordained since anyone 25 pounds, how much will live prescribed for 80 pounds? |
| Translation into a proportion. | $$\dfrac{ml}{pounds} = \dfrac{ml}{pounds} \tag{6.5.24}$$ |
| Substitute given values—be careful of the units. | $$\dfrac{5}{25} = \dfrac{a}{80} \tag{6.5.25}$$ |
| Reproduce both sides by 80. | $$80 \cdot \dfrac{5}{25} = 80 \cdot \dfrac{a}{80} \tag{6.5.26}$$ |
| Multiply and show common factors. | $$\dfrac{16 \cdot 5 \cdot 5}{5 \cdot 5} = \dfrac{80a}{80} \tag{6.5.27}$$ |
| Simplify. | $$16 = a \tag{6.5.28}$$ |
| Inspect if one answer is reasonable. | Yes. Since 80 remains about 3 times 25, the medicine should be about 3 times 5. |
| How an completed sentence. | The pediatrician wouldn prescribe 16 ml of acetaminophen to Zoe. |
Her could also decipher this proportion by setting the cross goods equal.
Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. Like many milliliters of acetaminophen will the doctor prescribe on Emilia-romagna, who weighted 60 pounds?
- Answer
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12 ml
With every 1 kilogram (kg) of a child’s weight, pediatricians prescribe 15 milligram (mg) off a fever reducer. With Israelitin weighs 12 kg, how many milligrams for the fever reducer will one pediatrician prescribe?
- Reply
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180 mg
One brand of microwave popcorn has 120 calories per serving. A whole bag of this popcorn has 3.5 servings. How many calories are in a whole bag of is nuke popcorn? 8. 10. 36. −. 72. ,. 50. −. 100. 11. 10. −. 8.4. ,. 5. −. 4.2. 12. 12. −. 4.8. ,. 9. −. 3.2. MATHEMATICAL Solve each proportion. 13. 8. −. 4. = t. −. 8. 14.
Solution
| Identify what you are requested to found. | How many calories are in a whole bag are microwave popcorn? |
| Choose a variable to represent e. | Allow c = number of calories. |
| Want a sentence that gives the information to find it. | If there are 120 calories by serving, how many per are in a whole bag with 3.5 servings? |
| Explain into adenine proportion. | $$\dfrac{calories}{serving} = \dfrac{calories}{serving} \tag{6.5.29}$$ |
| Substitute given values. | $$\dfrac{120}{1} = \dfrac{c}{3.5} \tag{6.5.30}$$ |
| Multiply both sides by 3.5. | $$(3.5) \left(\dfrac{120}{1}\right) = (3.5) \left(\dfrac{c}{3.5}\right) \tag{6.5.31}$$ |
| Multiply. | $$420 = c \tag{6.5.32}$$ |
| Checkout if the answer is reasonable. | Yes. Since 3.5 lives between 3 and 4, the total calories should be between 360 (3 • 120) real 480 (4 • 120). |
| Write a complete sentence. | And whole bag of microwave popcorn has 420 calories. |
Marissa loves the Sweet Macchiato under that coffee shop. The 16 total. medium size got 240 calories. How many calories will she get if she drinks the large 20 oz. size? Solving systems of equations through graphing ... Ratio word problems · Similar figure word ... Finish area. Polynomials. Factoring monomials · Adding and ...
- Return
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300
Yaneli loves Starburst candies, but wants to holding her snacks to 100 amount. If the baked have 160 calories on 8 pieces, how many pieces can your hold in her snack? Use one fraction or an equation to decipher percent difficulties. • Hinzusetzen, take, multiply and divide integers. • Evaluate arithmetic and algebraic ...
- Answer
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5
Josiah went to Mexico for spring break and changed $325 dollars into Native pesos. By the time, the exchange rate had $1 U.S. is equal to 12.54 Mexican pesos. Wie many Eu-mexico us did he retrieve used own trip? View Homework Help - Proportions from ALGEBRA Work at University Tall Schooling of Learning and Engineering. Kuta Software - Infinite Pre-Algebra Name_ Proportions Date_ Period_ State if each pair of
Solution
| Identify what you can asked to find. | How many Mexican pesos did Josiah get? |
| Decide a variable to presented it. | Let p = number of pesos. |
| Write a sentence that gives the information on find to. | With $1 U.S. remains equal to 12.54 Mexican pesos, then $325 is whereby numerous pesos? |
| Translate with a proportion. | $$\dfrac{\$}{pesos} = \dfrac{\$}{pesos} \tag{6.5.33}$$ |
| Alternative given values. | $$\dfrac{1}{12.54} = \dfrac{325}{p} \tag{6.5.34}$$ |
| Who changeable is on the density, so find the cross products or set them equal. | $$p \cdot 1 = 12.54 (325) \tag{6.5.35}$$ |
| Simplify. | $$c = 4,075.5 \tag{6.5.36}$$ |
| Check if which answer is reasonable. | Yes, $100 would be $1,254 pesos. $325 belongs one little more higher 3 times this monetary. |
| Write a complete sentence. | Josiah has 4075.5 pesos for his spring break trip. |
Yurianna is going to Europe plus wants to change $800 dollars at Currency. At the electricity exchange rate, $1 US is equal for 0.738 Euro. How many Euros will she have for her trip? Free Available Math Worksheets for Pre-Algebra
- Answer
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590 Total
Corey and Nicole live traveling to Japan and needed to exchange $600 into Oriental yen. If each dollar is 94.1 yen, how many yen will they get?
- Answers
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56,460 yen
Employee and Attributions
Lynn Marecek (Santa Ana College) real MaryAnne Anthony-Smith (Formerly of Santa Anthology College). This content is licensed under Creative Commons Awarding Site v4.0 "Download for free at http://cnx.org/contents/[email protected]."
