what is the solution to the proportion 5/9=m/63
half-dozen.v: Solve Proportions and their Applications (Part 1)
- Folio ID
- 5030
Skills to Develop
- Use the definition of proportion
- Solve proportions
- Solve applications using proportions
- Write percent equations as proportions
- Translate and solve percent proportions
be prepared!
Before you get started, take this readiness quiz.
- Simplify: \(\dfrac{\dfrac{1}{iii}}{4}\). If y'all missed this trouble, review Instance 4.5.viii.
- Solve: \(\dfrac{ten}{iv}\) = 20. If y'all missed this problem, review Example 4.12.5.
- Write equally a rate: Sale rode his bike 24 miles in 2 hours. If y'all missed this problem, review Case v.x.6.
Utilize the Definition of Proportion
In the section on Ratios and Rates nosotros saw some ways they are used in our daily lives. When two ratios or rates are equal, the equation relating them is called a proportion.
Definition: proportion
A proportion is an equation of the form \(\dfrac{a}{b} = \dfrac{c}{d}\), where b ≠ 0, d ≠ 0.
The proportion states two ratios or rates are equal. The proportion is read "a is to b, as c is to d".
The equation \(\dfrac{1}{2} = \dfrac{iv}{8}\) is a proportion because the two fractions are equal. The proportion \(\dfrac{1}{2} = \dfrac{iv}{8}\) is read "i is to two as 4 is to 8".
If we compare quantities with units, nosotros accept to be sure nosotros are comparing them in the right social club. For example, in the proportion \(\dfrac{20\; students}{1\; teacher} = \dfrac{60\; students}{3\; teachers}\) we compare the number of students to the number of teachers. We put students in the numerators and teachers in the denominators.
Example \(\PageIndex{1}\):
Write each sentence equally a proportion: (a) 3 is to vii as fifteen is to 35. (b) five hits in 8 at bats is the same as thirty hits in 48 at-bats. (c) $1.50 for 6 ounces is equivalent to $2.25 for 9 ounces.
Solution
(a) three is to 7 every bit 15 is to 35
| Write as a proportion. | $$\dfrac{three}{7} = \dfrac{15}{35}$$ |
(b) 5 hits in viii at bats is the same as 30 hits in 48 at-bats
| Write each fraction to compare hits to at-bats. | $$\dfrac{hits}{at-bats} = \dfrac{hits}{at-bats}$$ |
| Write 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 compare dollars to ounces. | $$\dfrac{\$}{ounces} = \dfrac{\$}{ounces}$$ |
| Write as a proportion. | $$\dfrac{1.50}{6} = \dfrac{2.25}{9}$$ |
Exercise \(\PageIndex{1}\):
Write each sentence as a proportion: (a) 5 is to ix as twenty is to 36. (b) 7 hits in 11 at-bats is the same as 28 hits in 44 at-bats. (c) $ii.50 for 8 ounces is equivalent to $3.75 for 12 ounces.
- Reply a
-
\(\frac{five}{ix} = \frac{twenty}{36}\)
- Answer b
-
\(\frac{7}{11} = \frac{28}{44}\)
- Answer c
-
\(\frac{2.50}{8} = \frac{3.75}{12}\)
Exercise \(\PageIndex{2}\):
Write each sentence as a proportion: (a) 6 is to 7 as 36 is to 42. (b) 8 adults for 36 children is the same every bit 12 adults for 54 children. (c) $3.75 for 6 ounces is equivalent to $2.50 for 4 ounces.
- Answer a
-
\(\frac{6}{7} = \frac{36}{42}\)
- Answer b
-
\(\frac{8}{36} = \frac{12}{54}\)
- Answer c
-
\(\frac{three.75}{6} = \frac{two.fifty}{4}\)
Look at the proportions \(\dfrac{1}{2} = \dfrac{4}{8}\) and \(\dfrac{2}{3} = \dfrac{vi}{9}\). From our piece of work with equivalent fractions we know these equations are true. But how do we know if an equation is a proportion with equivalent fractions if it contains fractions with larger numbers? To determine if a proportion is true, we notice the cross products of each proportion. To find the cross products, we multiply each denominator with the opposite numerator (diagonally across the equal sign). The results are chosen a cross products considering of the cross formed. The cross products of a proportion are equal.
Definition: Cross Products of a Proportion
For any proportion of the form \(\dfrac{a}{b} = \dfrac{c}{d}\), where b ≠ 0, d ≠ 0, its cantankerous products are equal.
Cross products can exist used to examination whether a proportion is true. To exam whether an equation makes a proportion, we notice the cross products. If they are the equal, we have a proportion.
Case \(\PageIndex{2}\):
Make up one's mind whether each equation is a proportion: (a) \(\dfrac{iv}{ix} = \dfrac{12}{28}\) (b) \(\dfrac{17.5}{37.5} = \dfrac{7}{15}\)
Solution
To make up one's mind if the equation is a proportion, nosotros find the cantankerous products. If they are equal, the equation is a proportion.
(a) \(\dfrac{four}{nine} = \dfrac{12}{28}\)
| Find the cantankerous products. | \[28 \cdot iv = 112 \qquad ix \cdot 12 = 108\] |
Since the cantankerous products are not equal, 28 · 4 ≠ 9 · 12, the equation is not a proportion.
(b) \(\dfrac{17.five}{37.5} = \dfrac{7}{15}\)
| Observe the cross products. | \[15 \cdot 17.5 = 262.v \qquad 37.5 \cdot seven = 262.5\] |
Since the cross products are equal, 15 • 17.5 = 37.five • seven, the equation is a proportion.
Exercise \(\PageIndex{3}\):
Make up one's mind whether each equation is a proportion: (a) \(\dfrac{7}{9} = \dfrac{54}{72}\) (b) \(\dfrac{24.5}{45.5} = \dfrac{7}{13}\)
- Respond a
-
no
- Answer b
-
yes
Exercise \(\PageIndex{4}\):
Decide whether each equation is a proportion: (a) \(\dfrac{eight}{ix} = \dfrac{56}{73}\) (b) \(\dfrac{28.five}{52.5} = \dfrac{8}{xv}\)
- Answer a
-
no
- Answer b
-
no
Solve Proportions
To solve a proportion containing a variable, nosotros retrieve that the proportion is an equation. All of the techniques we have used so far to solve equations even so use. In the next example, we will solve a proportion by multiplying by the Least Mutual Denominator (LCD) using the Multiplication Property of Equality.
Example \(\PageIndex{3}\):
Solve: \(\dfrac{x}{63} =\dfrac{4}{7}\).
Solution
| To isolate 10, multiply both sides by the LCD, 63. | $$\textcolor{red}{63} \left(\dfrac{x}{63}\right) = \textcolor{crimson}{63} \left(\dfrac{4}{7}\correct)$$ |
| Simplify. | $$x = \dfrac{9 \cdot \cancel{7} \cdot 4}{\abolish{vii}}$$ |
| Separate the mutual factors. | $$ten = 36$$ |
Check: To check our respond, we substitute into the original proportion.
| Substitute x = \(\textcolor{red}{36}\) | $$\dfrac{\textcolor{cerise}{36}}{63} \stackrel{?}{=} \dfrac{4}{seven}$$ |
| Show common factors. | $$\dfrac{iv \cdot ix}{vii \cdot nine} \stackrel{?}{=} \dfrac{4}{7}$$ |
| Simplify. | $$\dfrac{4}{7} = \dfrac{4}{7} \; \checkmark$$ |
Exercise \(\PageIndex{5}\):
Solve the proportion: \(\dfrac{n}{84} = \dfrac{11}{12}\).
- Answer
-
77
Exercise \(\PageIndex{half-dozen}\):
Solve the proportion: \(\dfrac{y}{96} = \dfrac{13}{12}\).
- Answer
-
104
When the variable is in a denominator, we'll employ the fact that the cross products of a proportion are equal to solve the proportions.
We tin can find the cross products of the proportion and so set them equal. And so we solve the resulting equation using our familiar techniques.
Example \(\PageIndex{four}\):
Solve: \(\dfrac{144}{a} =\dfrac{ix}{4}\).
Solution
Detect that the variable is in the denominator, and then nosotros will solve by finding the cross products and setting them equal.
| Discover the cross products and prepare them equal. | four • 144 = a • 9 |
| Simplify. | 576 = 9a |
| Separate both sides by 9. | $$\dfrac{576}{9} = \dfrac{9a}{9}$$ |
| Simplify. | $$64 = a$$ |
Bank check your reply.
| Substitute a = \(\textcolor{red}{64}\) | $$\dfrac{144}{\textcolor{ruby}{64}} \stackrel{?}{=} \dfrac{9}{4}$$ |
| Bear witness mutual factors. | $$\dfrac{9 \cdot xvi}{4 \cdot 16} \stackrel{?}{=} \dfrac{9}{4}$$ |
| Simplify. | $$\dfrac{9}{4} = \dfrac{nine}{4} \; \checkmark$$ |
Another method to solve this would be to multiply both sides by the LCD, 4a. Try it and verify that you get the aforementioned solution.
Exercise \(\PageIndex{vii}\):
Solve the proportion: \(\dfrac{91}{b} = \dfrac{seven}{5}\).
- Respond
-
65
Practise \(\PageIndex{8}\):
Solve the proportion: \(\dfrac{39}{c} = \dfrac{13}{8}\).
- Answer
-
24
Instance \(\PageIndex{5}\):
Solve: \(\dfrac{52}{91} = \dfrac{-4}{y}\)
Solution
| Find the cantankerous products and prepare them equal. |  |
| y • 52 = 91(-4) | |
| Simplify. | 52y = -364 |
| Divide both sides by 52. | $$\dfrac{52y}{52} = \dfrac{-364}{52}$$ |
| Simplify. | $$y = -7$$ |
Check:
| Substitute y = \(\textcolor{red}{-7}\) | $$\dfrac{52}{91} \stackrel{?}{=} \dfrac{-iv}{\textcolor{red}{-seven}}$$ |
| Prove common factors. | $$\dfrac{13 \cdot 4}{13 \cdot 7} \stackrel{?}{=} \dfrac{-four}{\textcolor{red}{-seven}}$$ |
| Simplify. | $$\dfrac{4}{vii} = \dfrac{4}{7} \; \checkmark$$ |
Practice \(\PageIndex{ix}\):
Solve the proportion: \(\dfrac{84}{98} = \dfrac{-half dozen}{x}\).
- Respond
-
-7
Practise \(\PageIndex{10}\):
Solve the proportion: \(\dfrac{-7}{y} = \dfrac{105}{135}\).
- Answer
-
-ix
Solve Applications Using Proportions
The strategy for solving applications that we have used before in this chapter, besides works for proportions, since proportions are equations. When we set up the proportion, nosotros must make sure the units are correct—the units in the numerators match and the units in the denominators friction match.
Case \(\PageIndex{6}\):
When pediatricians prescribe acetaminophen to children, they prescribe five milliliters (ml) of acetaminophen for every 25 pounds of the child'southward weight. If Zoe weighs lxxx pounds, how many milliliters of acetaminophen will her doctor prescribe?
Solution
| Identify what you lot are asked to discover. | How many ml of acetaminophen the doc will prescribe? |
| Choose a variable to correspond information technology. | Let a = ml of acetaminophen. |
| Write a sentence that gives the information to observe information technology. | If v ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds? |
| Translate into a proportion. | $$\dfrac{ml}{pounds} = \dfrac{ml}{pounds} \tag{vi.5.24}$$ |
| Substitute given values—be careful of the units. | $$\dfrac{five}{25} = \dfrac{a}{lxxx} \tag{half dozen.5.25}$$ |
| Multiply both sides by lxxx. | $$fourscore \cdot \dfrac{5}{25} = lxxx \cdot \dfrac{a}{80} \tag{6.five.26}$$ |
| Multiply and show common factors. | $$\dfrac{16 \cdot 5 \cdot 5}{5 \cdot 5} = \dfrac{80a}{fourscore} \tag{half-dozen.five.27}$$ |
| Simplify. | $$16 = a \tag{6.five.28}$$ |
| Check if the answer is reasonable. | Aye. Since eighty is nigh 3 times 25, the medicine should exist about three times 5. |
| Write a complete sentence. | The pediatrician would prescribe 16 ml of acetaminophen to Zoe. |
You lot could as well solve this proportion by setting the cantankerous products equal.
Exercise \(\PageIndex{11}\):
Pediatricians prescribe five milliliters (ml) of acetaminophen for every 25 pounds of a kid's weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs 60 pounds?
- Respond
-
12 ml
Practise \(\PageIndex{12}\):
For every 1 kilogram (kg) of a child's weight, pediatricians prescribe 15 milligrams (mg) of a fever reducer. If Isabella weighs 12 kg, how many milligrams of the fever reducer will the pediatrician prescribe?
- Answer
-
180 mg
Case \(\PageIndex{7}\):
I brand of microwave popcorn has 120 calories per serving. A whole handbag of this popcorn has three.5 servings. How many calories are in a whole bag of this microwave popcorn?
Solution
| Place what you are asked to discover. | How many calories are in a whole bag of microwave popcorn? |
| Choose a variable to represent information technology. | Let c = number of calories. |
| Write a sentence that gives the information to detect it. | If at that place are 120 calories per serving, how many calories are in a whole pocketbook with three.5 servings? |
| Translate into a proportion. | $$\dfrac{calories}{serving} = \dfrac{calories}{serving} \tag{half dozen.v.29}$$ |
| Substitute given values. | $$\dfrac{120}{i} = \dfrac{c}{3.5} \tag{6.v.30}$$ |
| Multiply both sides by 3.v. | $$(3.5) \left(\dfrac{120}{1}\right) = (3.v) \left(\dfrac{c}{three.v}\right) \tag{6.5.31}$$ |
| Multiply. | $$420 = c \tag{six.v.32}$$ |
| Check if the answer is reasonable. | Yes. Since 3.5 is betwixt iii and iv, the total calories should exist between 360 (3 • 120) and 480 (4 • 120). |
| Write a complete sentence. | The whole bag of microwave popcorn has 420 calories. |
Practise \(\PageIndex{13}\):
Marissa loves the Caramel Macchiato at the coffee shop. The sixteen oz. medium size has 240 calories. How many calories volition she become if she drinks the big xx oz. size?
- Answer
-
300
Exercise \(\PageIndex{xiv}\):
Yaneli loves Starburst candies, but wants to keep her snacks to 100 calories. If the candies accept 160 calories for viii pieces, how many pieces tin can she take in her snack?
- Answer
-
5
Example \(\PageIndex{8}\):
Josiah went to Mexico for spring break and inverse $325 dollars into Mexican pesos. At that time, the exchange charge per unit had $1 U.S. is equal to 12.54 Mexican pesos. How many Mexican pesos did he get for his trip?
Solution
| Identify what you are asked to notice. | How many Mexican pesos did Josiah get? |
| Choose a variable to correspond it. | Allow p = number of pesos. |
| Write a sentence that gives the information to find it. | If $ane U.Due south. is equal to 12.54 Mexican pesos, and so $325 is how many pesos? |
| Translate into a proportion. | $$\dfrac{\$}{pesos} = \dfrac{\$}{pesos} \tag{vi.v.33}$$ |
| Substitute given values. | $$\dfrac{1}{12.54} = \dfrac{325}{p} \tag{6.5.34}$$ |
| The variable is in the denominator, so find the cantankerous products and set them equal. | $$p \cdot one = 12.54 (325) \tag{vi.five.35}$$ |
| Simplify. | $$c = four,075.5 \tag{6.5.36}$$ |
| Check if the respond is reasonable. | Aye, $100 would exist $1,254 pesos. $325 is a little more than 3 times this amount. |
| Write a consummate sentence. | Josiah has 4075.5 pesos for his leap suspension trip. |
Exercise \(\PageIndex{15}\):
Yurianna is going to Europe and wants to change $800 dollars into Euros. At the electric current exchange rate, $one US is equal to 0.738 Euro. How many Euros will she have for her trip?
- Answer
-
590 Euros
Exercise \(\PageIndex{16}\):
Corey and Nicole are traveling to Japan and need to substitution $600 into Japanese yen. If each dollar is 94.ane yen, how many yen will they get?
- Answer
-
56,460 yen
Contributors and Attributions
-
Lynn Marecek (Santa Ana College) and MaryAnne Anthony-Smith (Formerly of Santa Ana College). This content is licensed nether Creative Commons Attribution License v4.0 "Download for free at http://cnx.org/contents/fd53eae1-fa2...49835c3c@v.191."
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Source: https://math.libretexts.org/Bookshelves/PreAlgebra/Book%3A_Prealgebra_(OpenStax)/06%3A_Percents/6.05%3A_Solve_Proportions_and_their_Applications_(Part_1)
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