Proportions - Algebra 1

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Question

The probability of an event is 3/11. Find the chance the event will not occur.

Answer

If the chance an event will happen is 3/11, that means there are 8 instances where the even would not occur out of every 11, giving us 8/11.

3 chances out of 11 = event

11 – 3 = 8

8 chances out of 11 = no event

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Question

Solve for :

$\small \frac{4}{7} = \frac{x}{28}$

Answer

Cross multiply:

$\small 7x\ =\ 4\times 28\ =112$

Solve for :

$\small 7x\ =\ 112$

$\frac{112}{7}=16$

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Question

Solve for in the following proportion statement:

$\small \frac{n}{4\frac{1}{2}} = \frac{4\frac{2}{3}}{2\frac{1}{4}}$

Answer

Use cross-multiplication to set up the following equation:

$\small \frac{n}{4\frac{1}{2}} = \frac{4\frac{2}{3}}{2\frac{1}{4}}$

$\small 2\tfrac{1}{4} \cdot n = 4\tfrac{1}{2} \cdot 4\tfrac{2}{3}$

Rewrite with improper fractions and solve:

$\small \small \frac{9}{4} \cdot n = \frac{9}{2} \cdot \frac{14}{3} = \frac{3}{1} \cdot \frac{7}{1} = \frac{21}{1}$

$\small n= \frac{21}{1} \div \frac{9}{4} = \frac{21}{1} \cdot \frac{4}{9} = \frac{7}{1} \cdot \frac{4}{3} = \frac{28}{3} = 9 \frac{1}{3}$

So the correct choice is $\small 9 \frac{1}{3}$ .

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Question

A car uses up gallons of gas in miles. Which expression best represents the distance, in kilometers, that the car can travel on 15 gallons of gas?

(1 mile is equivalent to 1.6 kilometers.)

Answer

gallons will allow the car to travel miles, or kilometers.

Set up a proportion, where is the distance:

$\small \frac{d}{15} = \frac{1.6y}{x}$

Solve for :

$\small d = \frac{1.6y}{x} \cdot 15$

$d = \frac{24y}{x}$

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Question

If Max drives 10 miles in 24 minutes, how many minutes will it take for him to drive 25 miles?

Answer

Set up a proportion, with miles driven on top and the time (in minutes) on the bottom:

$\frac{10}{24} = \frac{25}{x}$

Then, cross multiply:

, or

Finally, dividing both sides by 10 gives:

minutes.

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Question

$\textup{A taste test shows that out of every 5 people who participated, 3 of them preferred}$

$\textup{OK Cola to ABC Root Beer. If 100 people participated, how many preferred OK Cola?}$

Answer

$\textup{Set up an equation with the two proportions: }\frac{3}{5}=\frac{x}{100}$

$\textup{Cross multiply and solve: }5x=300;;;;;;;;;x=60$

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Question

If $\frac{3}{x} = \frac{x+4}{20}$ , what is the value of ?

Answer

Proportions are useful for solving many types of problems, but here our equation itself is a proportion.

To solve, we cross-multiply: the numerator of one side times the denominator of the other and vice versa. We are left with

$x^{2}+4x=60$ or

$x^{2}+4x-60=0$

A simple FOIL gives us

so our solutions are and .

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Question

$\textup{A bag contains 60 marbles, all of which are either red, white, or blue.}$

$\textup{If }\frac{2}{5}\textup{ of the marbles are red and }\frac{1}{4}\textup{ of the marbles are white,}$

$\textup{how many blue marbles are in the bag?}$

Answer

$\textup{Probability}= \frac{\textup{desired outcomes}}{\textup{total outcomes}}$

$\textup{ First find out how many red and white marbles there are.}$

$\textup{Red: }\frac{2}{5}=\frac{r}{60};;;5r=120;;;r=24;;;$

$\textup{White: }\frac{1}{4}=\frac{w}{60};;;4w=60;;;w=15$

$\textup{Blue: }b = 60-24-15;;;;;b=21$

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Question

A recipe for making 24 cookies calls for 6 cups of sugar. How many cups of sugar would be needed to make 16 cookies?

Answer

To solve the equation, you can set up a proportion with as the cups of sugar needed for making cookies. In this case, we know that it takes 6 cups of sugar to make 24 cookies, so we can set up the proportion as $\frac{x}{y}=\frac{6}{24}$ . Since we know that we need to make 16 cookies, we can substitute 16 for , $\frac{x}{16}=\frac{6}{24}$ . Cross multiply the fractions to get . Now, solve for to get a solution of 4.

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Question

To make 36 cookies, a recipe calls for 5 ounces of chocolate chips. How many ounces of chocolate chips would you need to make 900 cookies?

Answer

To figure out how many ounces of chocolate chips are needed to make 900 cookies, you simply need to set up a proportion. In this case, we know that 5 ounces of chocolate chips are needed to make 36 cookies, and we are trying to make a total of 900 cookies. Therefore, you can set up the proportion as

$\frac{5}{36}=\frac{x}{900}$

where is the ounces of chocolate chips needed to make 900 cookies.

For this proportion, solving for would give you a result of 125 ounces of chocolate chips.

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Question

If , what is the value of ?

Answer

To solve this equation, we need to set up a simple proportion. Since the variables and are already in use, let's call the quantity that we are solving for . From the given information, we know we can use the proportion $\frac{9x}{17y} = \frac{24x}{z}$ . Cross-multiplication yields . Dividing by the common term of and simplifying the right side gives us , so our solution must be .

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Question

Find .

$\frac{2}{9}=\frac{x}{36}$

Answer

This is a proportion problem. To find , we can use the cross-multiply method. This methods works by, first, multiplying the numerator of the first fraction with the denominator of the second fraction. Set this multiplication equal to the multiplication of the denominator of the first fraction with the numerator of the second fraction. In our problem, this is written as

divide by 9

$\frac{72}{9}=\frac{9x}{9}$

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Question

I have a bag full of blue and red marbles. If the ratio of blue to red marbles is 1:4 and if I have 16 blue ones, how many red marbles are in the bag.

Answer

First, translate the problem into a proportion equation. We have the ratio 1:4. Therefore, we have the fraction $\frac{1}{4}$ . Set that fraction equal to the number of blue (which is 16) marbles divide by the number of red marbles.

$\frac{1}{4}=\frac{16}{r}$

We need to find , the number of red marbles. Perform the cross multiplication

So there are 64 red marbles.

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Question

If , what is the value of ?

Answer

For this problem, we can divide both sides of the original equation by to obtain our answer. Another way we can solve this problem, however, is by using a proportion,

$\frac{3x}{141}=\frac{-x}{y}$

Cross-multiplication yields the equation

Dividing both sides by gives us

an equation that can be easily solved to get

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Question

Solve:

$\frac{(x+3)}{4}=\frac{7}{16}$

Answer

Multiply diagonally so that you get .

Isolate for and you get $-\frac{20}{16}$ .

Simplify and you get $-\frac{5}{4}$ .

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Question

$\frac{(9x+5)}{4}=\frac{2x}{8}$

Answer

Cross-multiply and you get

Isolate for and you get $-\frac{5}{8}$ .

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Question

Solve: $\frac{x}{2}= \frac{8}{x}$

Answer

To solve the proportion, cross multiply.

$x=\sqrt{16}$

$x=\pm 4$

The correct answer is $\pm4$ since both numbers satisfy the original problem.

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Question

Solve for .

$\frac{1}{2}=\frac{3}{x}$

Answer

Cross multiply.

$\frac{1}{2}=\frac{3}{x}$

We get

$\1\cdot x=2\cdot 3\ x=6$ .

That's the answer.

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Question

Solve for

$\frac{2}{5}=\frac{x}{3}$

Answer

Let's cross multiply.

$\frac{2}{5}=\frac{x}{3}$

We have

$\2\cdot 3=5\cdot x \6=5x$ .

Divide both sides by .

$x=\frac{6}{5}$ .

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Question

Solve for .

$\frac{3}{4}=\frac{x}{2}$

Answer

Let's cross multiply.

$\frac{3}{4}=\frac{x}{2}$

We have

$\4\cdot x=3\cdot 2\4x=6$ .

Divide both sides by , we get

$x=\frac{6}{4}=\frac{3}{2}$ .

The answer is reduced and we do that by dividing top and bottom by .

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