how to solve proportion word problems with fractions

Proportions Word Problems Worksheet: Answers and Explanations

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A proportion is a set of 2 fractions that equal each other. This article focuses on how to use proportions to solve real life problems.

Real World Uses of Proportions

• Modifying a budget for a restaurant chain that is expanding from 3 locations to 20 locations
• Creating a skyscraper from blueprints
• Calculating tips, commissions, and sales tax

Modifying a Recipe

On Monday, you are cooking enough white rice to serve exactly 3 people. The recipe calls for 2 cups of water and 1 cup of dry rice. On Sunday, you are going to serve rice to 12 people. How would the recipe change? If you’ve ever made rice, you know that this ratio—1 part dry rice and 2 parts water—is important. Mess it up, and you’ll be scooping a gummy mess on top of your guests' crawfish étouffée.

Because you are quadrupling your guest list (3 people * 4 = 12 people), you must quadruple your recipe. Cook 8 cups of water and 4 cups of dry rice. These shifts in a recipe demonstrate the heart of proportions: using a ratio to accommodate life's greater and smaller changes.

Algebra and Proportions 1

Sure, with the right numbers, you can forgo setting up an algebraic equation to determine the amounts of dry rice and water. What happens when the numbers are not so friendly, however? On Thanksgiving, you'll be serving rice to 25 people. How much water do you need?

Because the ratio of 2 parts water and 1 part dry rice applies to cooking 25 servings of rice, use a proportion to determine the quantity of ingredients.

Note : Translating a word problem into an equation is super important. Yes, you can solve an incorrectly set up equation and find an answer. You can also mix rice and water together to create "food" to serve at Thanksgiving. Whether the answer or food is palatable depends on the equation.

Think about what you know:

• 3 servings of cooked rice = 2 cups of water; 1 cup of dry rice 25 servings of cooked rice = ? cups of water; ? cup of dry rice
• 3 servings of cooked rice/25 servings of cooked rice = 2 cups of water/ x cups of water
• 3/25 = 2/ x

Cross multiply. Hint : Write these fractions vertically to get the full understanding of cross multiplying. To cross multiply, take the first fraction's numerator and multiply it by the second fraction's denominator. Then take the second fraction's numerator and multiply it by the first fraction's denominator.

3 * x = 2 * 25 3 x = 50 Divide both sides of the equation by 3 to solve for x . 3 x /3 = 50/3 x = 16.6667 cups of water Freeze- verify that the answer is correct. Is 3/25 = 2/16.6667? 3/25 = .12 2/16.6667= .12 Whoo hoo! The answer 16.6667 cups of water is correct. 

Ratio and Proportions Word Problem 1: The Brownie Recipe

Damian is making brownies to serve at the family picnic. If the recipe calls for 2 ½ cups of cocoa to serve 4 people, how many cups will he need if there will be 60 people at the picnic? 37.5 cups

What do you know? 2 ½ cups = 4 people ? cups = 60 people 2 ½ cups/ x cups = 4 people/60 people 2 ½/ x = 4/60 Cross Multiply. 2 ½ * 60 = 4 * x 150 = 4 x Divide both sides by 4 to solve for x . 150/4 = 4 x /4 37.5 = x 37.5 cups

Use common sense to verify that the answer is correct. The initial recipe serves 4 people and is modified to serve 60 people. Of course, the new recipe has to serve 15 times more people. Therefore, the amount of cocoa has to be multiplied by 15. Is 2 ½ * 15 = 37.5? Yes.

Ratio and Proportions Word Problem 2: Growing Little Piglets

A piglet can gain 3 pounds in 36 hours. If this rate continues, the pig will reach 18 pounds in  216  hours.

What do you know? 3 pounds = 36 hours 18 pounds = ? hours 3 pounds/18 pounds = 36 hours/ ? hours 3/18 = 36/ x

Cross Multiply. 3 * x = 36 * 18 3 x = 648

Divide both sides by 3 to solve for x . 3 x /3 = 648/3 x = 216 216 hours

Use common sense to verify that the answer is correct. A piglet can gain 3 pounds in 36 hours, which is a rate of 1 pound for every 12 hours. That means that for every pound a piglet gains, 12 hours will pass. Therefore 18 *12, or 216 pounds, is the correct answer.

Ratio and Proportions Word Problem 3: The Hungry Rabbit

Denise’s rabbit can eat 70 pounds of food in 80 days. How long will it take the rabbit to eat 87.5 pounds? 100 days

What do you know? 70 pounds = 80 days 87.5 pounds = ? days 70 pounds/87.5 pounds = 80 days/ x days 70/87.5 = 80/ x

Cross Multiply. 70 * x = 80 * 87.5 70 x = 7000

Divide both sides by 70 to solve for x . 70 x /70 = 7000/70 x = 100

Use Algebra to verify the answer. Is 70/87.5 = 80/100? 70/87.5 = .8 80/100 = .8

Ratio and Proportions Word Problem 4: The Long Road Trip

Jessica drives 130 miles every two hours. If this rate continues, how long will it take her to drive 1,000 miles? 15.38 hours

What do you know? 130 miles = 2 hours 1,000 miles = ? hours 130 miles/1,000 miles = 2 hours/? hours 130/1000 = 2/ x

Cross Multiply. 130 * x = 2 * 1000 130 x = 2000

Divide both sides of the equation by 130 to solve for x . 130 x /130 = 2000/130 x = 15.38 hours

Use Algebra to verify the answer. Does 130/1000 = 2/15.38? 130/1000 = .13 2/15.38 is approximately .13

• Proportions Word Problems Worksheet 1
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Solving Proportions: Word Problems

Ratios Proportions Proportionality Solving Word Problems Similar Figures Sun's Rays / Parts

Many "proportion" word problems can be solved using other methods, so they may be familiar to you. For instance, if you've learned about straight-line equations, then you've learned about the slope of a straight line, and how this slope is sometimes referred to as being "rise over run".

But that word "over" gives a hint that, yes, we're talking about a fraction. And this means that "rise over run" can be discussed within the context of proportions.

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You are installing rain gutters across the back of your house. The directions say that the gutters should decline katex.render("\\small{ \\bm{\\color{green}{ \\frac{1}{4} }}}", typed01); 1 / 4 inch for every four feet of lateral run. The gutters will be spanning thirty-seven feet. How much lower than the starting point (that is, how much lower than the high end) should the low end of the gutters be?

Rain gutters have to be slightly sloped so the rainwater will drain toward and then down the downspout. As I go from the high end of the guttering to the low end, for every four-foot length that I go sideways, the gutters should decline [be lower by] one-quarter inch. So how much must the guttering decline over the thirty-seven foot span? I'll set up the proportion, using " d " to stand for the distance I'm needing to find.

There is a variable in only one part of my proportion, so I can use the shortcut method to solve.

d = [(37)(1/4)]/4

For convenience sake (because my tape measure isn't marked in decimals), I'll convert this answer to mixed-number form:

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As is always the case with "solving" exercises, we can check our answers by plugging them back into the original problem. In this case, we can verify the size of the "drop" from one end of the house to the other by checking the products of the means and the extremes (that is, by confirming that the cross-multiplications match) of the completed proportion:

(1/4)/4 = 2.3125/37

Converting the "one-fourth" to " 0.25 ", we get:

(0.25)(37) = 9.25

(4)(2.3125) = 9.25

Since the values match, then the proportionality must have been solved correctly, and the solution must be right.

Biologists need to know roughly how many fish live in a certain lake, but they don't want to stress or otherwise harm the fish by draining or dragnetting the lake. Instead, they let down small nets in a few different spots around the lake, catching, tagging, and releasing 96 fish. A week later, after the tagged fish have had a chance to mix thoroughly with the general population, the biologists come back and let down their nets again. They catch 72 fish, of which 4 are tagged. Assuming that the catch is representative, how many fish live in the lake?

As far as I know, biologists and park managers actually use this technique for estimating populations. The idea is that, after allowing enough time (it is hoped) for the tagged fish to circulate throughout the lake, these fish will then be evenly mixed in with the total population. When the researchers catch some fish later, the ratio of tagged fish in the sample to untagged is representative of the ratio of the 96 fish that they tagged with the total population.

I'll use " f " to stand for the total number of fish in the lake, and set up my ratios with the numbers of "tagged" fish on top. Then I'll set up and solve the proportion:

Because the variable is in only one part of the proportion, I can use the shortcut method to solve.

f = [(96)(72)]/4

This tells me that the estimated population is:

about 1,728 fish

Another type of "proportion" word problem is unit conversion, which looks like this:

How many feet per second are equivalent to 60 mph?

To complete this exercise, I will need conversion factors, which are just ratios. (If you're doing this kind of problem, then you should have access — in your textbook or in a handout, for instance — to basic conversion factors. If not, then your instructor is probably expecting that you have these factors memorized.)

I'll set everything up in a long multiplication so that the units cancel :

88 feet per second

Take note of how I set up the conversion factors for my multiplicate (above) in not-necessarily-standard ways. For instance, one usually says "sixty minutes in an hour", not "one hour in sixty minutes". So why did I enter the hour-minute conversion factor (in the second line of my computations above) as "one hour per sixty minutes"?

Because doing so lined up the fractions so that the unit of "hours" in my conversion factor would cancel off with the "hours" in the original " 60 miles per hour". This cancelling-units thing is an important technique, and you should review it further if you are not comfortable with it.

A particular cookie recipe calls for 225 grams of flour for one batch of thirty cookies. Jade would like to make as many cookies as possible for the upcoming block party, and flour is his only constraint (he's got loads of sugar, eggs, etc). If he has 1.206 kilograms of flour, and assuming that all cookies are the same size, approximately how many cookies can he make? (Round to an appropriate whole number.)

I've got two elements here for my proportion: grams of flour and number of cookies. I got to "grams" first when reading the exercise, so I'll put "grams" on top in my proportion.

Since the relationship is given to me in terms of grams, not kilograms, I'll need to convert Jade's on-hand measure to " 1,206 grams, also. I'll use " c " to stand for the number that I'm trying to figure out for "cookies".

(grams)/(cookies): 225/30 = 1206/ c

Since I have an unknown in only one spot in this proportion, I can use the shortcut method to solve.

c = [(1206)(30)]/225

c = 36180/225

Ohhh! Now I see why the instructions said to round to an "appropriate" whole number: Jade can only make whole cookies; the "point-eight" of a cookie will be an undersized niblet that he'll eat before heading to the party.

While normally I'd round this number up to get my whole-number answer, in this case I need to round down ; in other words, in this context (namely, of all the cookies being the same size), I have to ignore the fractional portion (that is, the point-eight decimal part) to get the desired answer.

160 cookies

Kumar lives in Croatia, and is visiting relatives in India. The current exchange rate is one Euro ( €1 ) to 80.45 Indian rupees ( ₹80.45 ). He wants to buy a gift for them, which costs ₹3,759 . How many Euros will this gift cost him? (State your answer accurate to two decimal places.)

They've given me an exchange rate, which is, effectively, just another conversion factor, like the "miles per hour" exercise above. So I'll set up my proportion, with Euros on top, and will use e to stand for the number of Euros he'll need.

(Euros)/(Rupees): 1/80.45 = e /3759

I'll use the shortcut method to solve:

e = [(3759)(1)]/80.45

e = 46.72467371...

Rounding to two decimal places, Kumar will be spending:

€46.72

Other than for the rate-conversion exercise above, we've been able to solve all of the proportions by the shortcut method. You will likely find this to be the case in your homework, also. But it is always possible that you'll get a question where you'll be better off using cross-multiplication instead.

George has heard from two different sources about the pay range at a particular company. One source says that the ratio of lowest pay to highest pay is 3 : 7 . The other source says that the top earner annually makes about $57,000 more than the lowest earner. What are the approximate salaries for the highest and lowest earners? (Round to the nearest thousand.)

I know that the ratio is 3 : 7 , so I'll be using the fraction 3/7 for one side of my proportion. If the lowest pay rate, in thousands of dollars, is L , then the highest is L  + 57 . My proportion is:

(lowest)/(highest): 3/7 = L /( L + 57)

Because there are variables in two of the parts of this proportion, the shortcut method won't be as useful as cross-multiplication to clear all the fractions. So I'll cross-multiply:

3/7 = L /( L + 57)

3( L + 57) = 7 L

3 L + 171 = 7 L

Remembering that I dropped the trailing zeroes and am counting by thousands, the above number means that the lowest salary is (rounded to the nearest thousand) approximately $43,000 . Then the highest salary, being around $57K more, is approximately $100,000 .

lowest: $43,000 highest: $100,000

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Praxis Core Math

Course: praxis core math   >   unit 1.

• Rational number operations | Lesson
• Rational number operations | Worked example

Ratios and proportions | Lesson

• Ratios and proportions | Worked example
• Percentages | Lesson
• Percentages | Worked example
• Rates | Lesson
• Rates | Worked example
• Naming and ordering numbers | Lesson
• Naming and ordering numbers | Worked example
• Number concepts | Lesson
• Number concepts | Worked example
• Counterexamples | Lesson
• Counterexamples | Worked example
• Pre-algebra word problems | Lesson
• Pre-algebra word problems | Worked example
• Unit reasoning | Lesson
• Unit reasoning | Worked example

What are ratios and proportions?

What skills are tested.

• Identifying and writing equivalent ratios
• Solving word problems involving ratios
• Solving word problems using proportions

How do we write ratios?

• The ratio of lemon juice to sugar is a part-to-part ratio. It compares the amount of two ingredients.
• The ratio of lemon juice to lemonade is a part-to-whole ratio. It compares the amount of one ingredient to the sum of all ingredients.
• Determine whether the ratio is part to part or part to whole.
• Calculate the parts and the whole if needed.
• Plug values into the ratio.
• Simplify the ratio if needed. Integer-to-integer ratios are preferred.

How do we use proportions?

• Write an equation using equivalent ratios.
• Plug in known values and use a variable to represent the unknown quantity.
• If the numeric part of one ratio is a multiple of the corresponding part of the other ratio, we can calculate the unknown quantity by multiplying the other part of the given ratio by the same number.
• If the relationship between the two ratios is not obvious, solve for the unknown quantity by isolating the variable representing it.
• (Choice A)   1 : 4 1:4 1 : 4 1, colon, 4 A 1 : 4 1:4 1 : 4 1, colon, 4
• (Choice B)   1 : 2 1:2 1 : 2 1, colon, 2 B 1 : 2 1:2 1 : 2 1, colon, 2
• (Choice C)   1 : 1 1:1 1 : 1 1, colon, 1 C 1 : 1 1:1 1 : 1 1, colon, 1
• (Choice D)   2 : 1 2:1 2 : 1 2, colon, 1 D 2 : 1 2:1 2 : 1 2, colon, 1
• (Choice E)   4 : 1 4:1 4 : 1 4, colon, 1 E 4 : 1 4:1 4 : 1 4, colon, 1
• (Choice A)   1 6 \dfrac{1}{6} 6 1 ​ start fraction, 1, divided by, 6, end fraction A 1 6 \dfrac{1}{6} 6 1 ​ start fraction, 1, divided by, 6, end fraction
• (Choice B)   1 3 \dfrac{1}{3} 3 1 ​ start fraction, 1, divided by, 3, end fraction B 1 3 \dfrac{1}{3} 3 1 ​ start fraction, 1, divided by, 3, end fraction
• (Choice C)   2 5 \dfrac{2}{5} 5 2 ​ start fraction, 2, divided by, 5, end fraction C 2 5 \dfrac{2}{5} 5 2 ​ start fraction, 2, divided by, 5, end fraction
• (Choice D)   1 2 \dfrac{1}{2} 2 1 ​ start fraction, 1, divided by, 2, end fraction D 1 2 \dfrac{1}{2} 2 1 ​ start fraction, 1, divided by, 2, end fraction
• (Choice E)   2 3 \dfrac{2}{3} 3 2 ​ start fraction, 2, divided by, 3, end fraction E 2 3 \dfrac{2}{3} 3 2 ​ start fraction, 2, divided by, 3, end fraction
• Your answer should be
• an integer, like 6 6 6 6
• a simplified proper fraction, like 3 / 5 3/5 3 / 5 3, slash, 5
• a simplified improper fraction, like 7 / 4 7/4 7 / 4 7, slash, 4
• a mixed number, like 1   3 / 4 1\ 3/4 1   3 / 4 1, space, 3, slash, 4
• an exact decimal, like 0.75 0.75 0 . 7 5 0, point, 75
• a multiple of pi, like 12  pi 12\ \text{pi} 1 2   pi 12, space, start text, p, i, end text or 2 / 3  pi 2/3\ \text{pi} 2 / 3   pi 2, slash, 3, space, start text, p, i, end text

Things to remember

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