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Solve Proportion and Similar Figure Applications

Solve proportions

Solve Proportions

When two rational expressions are equal, the equation relating them is called a proportion.

The equation $\frac{1}{2}=\frac{4}{8}$ is a proportion because the two fractions are equal. The proportion $\frac{1}{2}=\frac{4}{8}$ is read “1 is to 2 as 4 is to 8.”

Proportions are used in many applications to ‘scale up’ quantities. We’ll start with a very simple example so you can see how proportions work. Even if you can figure out the answer to the example right away, make sure you also learn to solve it using proportions.

Suppose a school principal wants to have 1 teacher for 20 students. She could use proportions to find the number of teachers for 60 students. We let x be the number of teachers for 60 students and then set up the proportion:

\[\frac{1\ \text{teacher}}{20\ \text{students}}=\frac{x\ \text{teachers}}{60\ \text{students}}\]

We are careful to match the units of the numerators and the units of the denominators—teachers in the numerators, students in the denominators.

Since a proportion is an equation with rational expressions, we will solve proportions the same way we solved equations in Solve Rational Equations. We’ll multiply both sides of the equation by the LCD to clear the fractions and then solve the resulting equation.

So let’s finish solving the principal’s problem now. We will omit writing the units until the last step.


Multiply both sides by the LCD, 60.
Simplify.
	The principal needs 3 teachers for 60 students.

Example

Try it.

Solve the proportion: $\frac{x}{63}=\frac{4}{7}.$

Solution


To isolate $x$, multiply both sides by the LCD, 63.
Simplify.
Divide the common factors.
Check. To check our answer, we substitute into the original proportion.


Show common factors.
Simplify.

Example

Try it.

Solve the proportion: $\frac{144}{a}=\frac{9}{4}.$

Solution


Multiply both sides by the LCD.
Remove common factors on each side.
Simplify.
Divide both sides by 9.
Simplify.
Check.


Show common factors.
Simplify.

Example

Try it.

Solve the proportion: $\frac{n}{n+14}=\frac{5}{7}.$

Solution


Multiply both sides by the LCD.
Remove common factors on each side.
Simplify.
Solve for $n$.

Check.


Simplify.
Show common factors.
Simplify.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Solve Similar Figure Applications

When you shrink or enlarge a photo on a phone or tablet, figure out a distance on a map, or use a pattern to build a bookcase or sew a dress, you are working with similar figures. If two figures have exactly the same shape, but different sizes, they are said to be similar. One is a scale model of the other. All their corresponding angles have the same measures and their corresponding sides are in the same ratio.

For example, the two triangles in are similar. Each side of $\text{\Delta }ABC$ is 4 times the length of the corresponding side of $\text{\Delta }XYZ$.

This is summed up in the Property of Similar Triangles.

To solve applications with similar figures we will follow the Problem-Solving Strategy for Geometry Applications we used earlier.

The next example shows how similar triangles are used with maps.

We can use similar figures to find heights that we cannot directly measure.

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Key Concepts

Property of Similar Triangles
- If $\text{\Delta }ABC$ is similar to $\text{\Delta }XYZ$, then their corresponding angle measures are equal and their corresponding sides are in the same ratio.
Problem Solving Strategy for Geometry Applications
1. Read the problem and make sure all the words and ideas are understood. Draw the figure and label it with the given information.
2. Identify what we are looking for.
3. Name what we are looking for by choosing a variable to represent it.
4. Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information.
5. Solve the equation using good algebra techniques.
6. Check the answer in the problem and make sure it makes sense.
7. Answer the question with a complete sentence.

Solve Proportion and Similar Figure Applications

Solve Proportions

In the following exercises, solve.

Try it.

$\frac{x}{56}=\frac{7}{8}$

Solution

$49$

Try it.

$\frac{n}{91}=\frac{8}{13}$

Try it.

$\frac{49}{63}=\frac{z}{9}$

Solution

$7$

Try it.

$\frac{56}{72}=\frac{y}{9}$

Try it.

$\frac{5}{a}=\frac{65}{117}$

Solution

$9$

Try it.

$\frac{4}{b}=\frac{64}{144}$

Try it.

$\frac{98}{154}=\frac{-7}{p}$

Solution

$-11$

Try it.

$\frac{72}{156}=\frac{-6}{q}$

Try it.

$\frac{a}{-8}=\frac{-42}{48}$

Solution

$7$

Try it.

$\frac{b}{-7}=\frac{-30}{42}$

Try it.

$\frac{2.7}{j}=\frac{0.9}{0.2}$

Solution

$0.6$

Try it.

$\frac{2.8}{k}=\frac{2.1}{1.5}$

Try it.

$\frac{a}{a+12}=\frac{4}{7}$

Solution

$16$

Try it.

$\frac{b}{b-16}=\frac{11}{9}$

Try it.

$\frac{c}{c-104}=-\frac{5}{8}$

Solution

$40$

Try it.

$\frac{d}{d-48}=-\frac{13}{3}$

Try it.

$\frac{m+90}{25}=\frac{m+30}{15}$

Solution

$60$

Try it.

$\frac{n+10}{4}=\frac{40-n}{6}$

Try it.

$\frac{2p+4}{8}=\frac{p+18}{6}$

Solution

$30$

Try it.

$\frac{q-2}{2}=\frac{2q-7}{18}$

Try it.

Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Jocelyn, who weighs 45 pounds?

Solution

$9\ \text{ml}$

Try it.

Brianna, who weighs 6 kg, just received her shots and needs a pain killer. The pain killer is prescribed for children at 15 milligrams (mg) for every 1 kilogram (kg) of the child’s weight. How many milligrams will the doctor prescribe?

Try it.

A veterinarian prescribed Sunny, a 65 pound dog, an antibacterial medicine in case an infection emerges after her teeth were cleaned. If the dosage is 5 mg for every pound, how much medicine was Sunny given?

Solution

$325\ \text{mg}$

Try it.

Belle, a 13 pound cat, is suffering from joint pain. How much medicine should the veterinarian prescribe if the dosage is 1.8 mg per pound?

Try it.

A new energy drink advertises 106 calories for 8 ounces. How many calories are in 12 ounces of the drink?

Solution

$159\ \text{calories}$

Try it.

One 12 ounce can of soda has 150 calories. If Josiah drinks the big 32 ounce size from the local mini-mart, how many calories does he get?

Try it.

A new 7 ounce lemon ice drink is advertised for having only 140 calories. How many ounces could Sally drink if she wanted to drink just 100 calories?

Solution

$5\ \text{oz}$

Try it.

Reese loves to drink healthy green smoothies. A 16 ounce serving of smoothie has 170 calories. Reese drinks 24 ounces of these smoothies in one day. How many calories of smoothie is he consuming in one day?

Try it.

Janice is traveling to Canada and will change $250 US dollars into Canadian dollars. At the current exchange rate, $1 US is equal to $1.01 Canadian. How many Canadian dollars will she get for her trip?

Solution

$252.5\ \text{Canadian dollars}$

Try it.

Todd is traveling to Mexico and needs to exchange $450 into Mexican pesos. If each dollar is worth 12.29 pesos, how many pesos will he get for his trip?

Try it.

Steve changed $600 into 480 Euros. How many Euros did he receive for each US dollar?

Solution

$0.80\ \text{Euros}$

Try it.

Martha changed $350 US into 385 Australian dollars. How many Australian dollars did she receive for each US dollar?

Try it.

When traveling to Great Britain, Bethany exchanged her $900 into 570 British pounds. How many pounds did she receive for each American dollar?

Solution

$0.63\ \text{British pounds}$

Try it.

A missionary commissioned to South Africa had to exchange his $500 for the South African Rand which is worth 12.63 for every dollar. How many Rand did he have after the exchange?

Try it.

Ronald needs a morning breakfast drink that will give him at least 390 calories. Orange juice has 130 calories in one cup. How many cups does he need to drink to reach his calorie goal?

Solution

$3\ \text{cups}$

Try it.

Sarah drinks a 32-ounce energy drink containing 80 calories per 12 ounce. How many calories did she drink?

Try it.

Elizabeth is returning to the United States from Canada. She changes the remaining 300 Canadian dollars she has to $230.05 in American dollars. What was $1 worth in Canadian dollars?

Solution

$1.30\ \text{Canadian dollars}$

Try it.

Ben needs to convert $1000 to the Japanese Yen. One American dollar is worth 123.3 Yen. How much Yen will he have?

Try it.

A golden retriever weighing 85 pounds has diarrhea. His medicine is prescribed as 1 teaspoon per 5 pounds. How much medicine should he be given?

Solution

$17\ \text{tsp}$

Try it.

Five-year-old Lacy was stung by a bee. The dosage for the anti-itch liquid is 150 mg for her weight of 40 pounds. What is the dosage per pound?

Try it.

Karen eats $\frac{1}{2}$ cup of oatmeal that counts for 2 points on her weight loss program. Her husband, Joe, can have 3 points of oatmeal for breakfast. How much oatmeal can he have?

Solution

$\frac{3}{4}$ cup

Try it.

An oatmeal cookie recipe calls for $\frac{1}{2}$ cup of butter to make 4 dozen cookies. Hilda needs to make 10 dozen cookies for the bake sale. How many cups of butter will she need?

Solve Similar Figure Applications

In the following exercises, $\text{\Delta }ABC$ is similar to $\text{\Delta }XYZ$. Find the length of the indicated side.

Try it.

side b

Solution

$12$

Try it.

side x

In the following exercises, $\text{\Delta }DEF$ is similar to $\text{\Delta }NPQ$.

Try it.

Find the length of side d.

Solution

$\frac{77}{18}$

Try it.

Find the length of side q.

In the following two exercises, use the map shown. On the map, New York City, Chicago, and Memphis form a triangle whose sides are shown in the figure below. The actual distance from New York to Chicago is 800 miles.

Try it.

Find the actual distance from New York to Memphis.

Solution

950 miles

Try it.

Find the actual distance from Chicago to Memphis.

In the following two exercises, use the map shown. On the map, Atlanta, Miami, and New Orleans form a triangle whose sides are shown in the figure below. The actual distance from Atlanta to New Orleans is 420 miles.

Try it.

Find the actual distance from New Orleans to Miami.

Solution

680 miles

Try it.

Find the actual distance from Atlanta to Miami.

Try it.

A 2 foot tall dog casts a 3 foot shadow at the same time a cat casts a one foot shadow. How tall is the cat?

Solution

$\frac{2}{3}$ foot $(8\ \text{in})$

Try it.

Larry and Tom were standing next to each other in the backyard when Tom challenged Larry to guess how tall he was. Larry knew his own height is 6.5 feet and when they measured their shadows, Larry’s shadow was 8 feet and Tom’s was 7.75 feet long. What is Tom’s height?

Try it.

The tower portion of a windmill is 212 feet tall. A six foot tall person standing next to the tower casts a seven foot shadow. How long is the windmill’s shadow?

Solution

$247.3\ \text{feet}$

Try it.

The height of the Statue of Liberty is 305 feet. Nicole, who is standing next to the statue, casts a 6 foot shadow and she is 5 feet tall. How long should the shadow of the statue be?

Condensed — the full section is in OpenStax Elementary Algebra 2e.

Practice (40)

Try each one on paper first. Reveal the answer to check; verified ones can be opened in the solver for every step.

Solve $\frac{n}{3}=30$.
If you missed this problem, review .

ჲრკპთირვ ჲრდჲგჲპა.

$n=90$
The perimeter of a triangular window is 23 feet. The lengths of two sides are ten feet and six feet. How long is the third side?
If you missed this problem, review .

ჲრკპთირვ ჲრდჲგჲპა.

$7\ \text{feet}$
Solve the proportion: $\frac{x}{63}=\frac{4}{7}.$

ჲრკპთირვ ჲრდჲგჲპა.

To isolate $x$, multiply both sides by the LCD, 63.
Simplify.
Divide the common factors.
Check. To check our answer, we substitute into the original proportion.
Show common factors.
Simplify.
Solve the proportion: $\frac{n}{84}=\frac{11}{12}.$

ჲრკპთირვ ჲრდჲგჲპა.

$77$
Solve the proportion: $\frac{y}{96}=\frac{13}{12}.$

ჲრკპთირვ ჲრდჲგჲპა.

$104$
Solve the proportion: $\frac{144}{a}=\frac{9}{4}.$

ჲრკპთირვ ჲრდჲგჲპა.

Multiply both sides by the LCD.
Remove common factors on each side.
Simplify.
Divide both sides by 9.
Simplify.
Check.
Show common factors.
Simplify.
Solve the proportion: $\frac{91}{b}=\frac{7}{5}.$

ჲრკპთირვ ჲრდჲგჲპა.

$65$
Solve the proportion: $\frac{39}{c}=\frac{13}{8}.$

ჲრკპთირვ ჲრდჲგჲპა.

$24$
Solve the proportion: $\frac{n}{n+14}=\frac{5}{7}.$

ჲრკპთირვ ჲრდჲგჲპა.

Multiply both sides by the LCD.
Remove common factors on each side.
Simplify.
Solve for $n$.
Check.
Simplify.
Show common factors.
Simplify.
Solve the proportion: $\frac{y}{y+55}=\frac{3}{8}.$

ჲრკპთირვ ჲრდჲგჲპა.

$33$
Solve the proportion: $\frac{z}{z-84}=-\frac{1}{5}.$

ჲრკპთირვ ჲრდჲგჲპა.

$14$
Solve: $\frac{p+12}{9}=\frac{p-12}{6}.$

ჲრკპთირვ ჲრდჲგჲპა.

Multiply both sides by the LCD, 18.
Simplify.
Distribute.
Solve for $p$.
Check.
Simplify.
Divide.
Solve: $\frac{v+30}{8}=\frac{v+66}{12}.$

ჲრკპთირვ ჲრდჲგჲპა.

$42$
Solve: $\frac{2x+15}{9}=\frac{7x+3}{15}.$

ჲრკპთირვ ჲრდჲგჲპა.

$6$

When pediatricians prescribe acetaminophen to children, they prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of the child’s weight. If Zoe weighs 80 pounds, how many milliliters of acetaminophen will her doctor prescribe?

ჲრკპთირვ ჲრდჲგჲპა.

Identify what we are asked to find, and choose a variable to represent it.		How many ml of acetaminophen will the doctor prescribe?
		Let $a=\text{ml}$ of acetaminophen.
Write a sentence that gives the information to find it.		If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds?
Translate into a proportion–be careful of the units. $\frac{\text{ml}}{\text{pounds}}=\frac{\text{ml}}{\text{pounds}}$
Multiply both sides by the LCD, 400.
Remove common factors on each side.
Simplify, but don't multiply on the left. Notice what the next step will be.
Solve for $a$.
Check.
Is the answer reasonable?
Yes, since 80 is about 3 times 25, the medicine should be about 3 times 5. So 16 ml makes sense.

Write a complete sentence.		The pediatrician would prescribe 16 ml of acetaminophen to Zoe.

Pediatricians prescribe 5 milliliters (ml) of acetaminophen for every 25 pounds of a child’s weight. How many milliliters of acetaminophen will the doctor prescribe for Emilia, who weighs 60 pounds?

ჲრკპთირვ ჲრდჲგჲპა.

$12\ \text{ml}$
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?

ჲრკპთირვ ჲრდჲგჲპა.

$180\ \text{mg}$

A 16-ounce iced caramel macchiato has 230 calories. How many calories are there in a 24-ounce iced caramel macchiato?

ჲრკპთირვ ჲრდჲგჲპა.

Identify what we are asked to find, and choose a variable to represent it.		How many calories are in a 24 ounce iced caramel macchiato?
		Let $c=\text{calories}$ in 24 ounces.
Write a sentence that gives the information to find it.		If there are 230 calories in 16 ounces, then how many calories are in 24 ounces?
Translate into a proportion–be careful of the units. $\frac{\text{calories}}{\text{ounce}}=\frac{\text{calories}}{\text{ounce}}$
Multiply both sides by the LCD, 48.
Remove common factors on each side.
Simplify.
Solve for $c$.

Check.
Is the answer reasonable?
Yes, 345 calories for 24 ounces is more than 290 calories for 16 ounces, but not too much more.

Write a complete sentence.		There are 345 calories in a 24-ounce iced caramel macchiato.

At a fast-food restaurant, a 22-ounce chocolate shake has 850 calories. How many calories are in their 12-ounce chocolate shake? Round your answer to nearest whole number.

ჲრკპთირვ ჲრდჲგჲპა.

$464\ \text{calories}$
Yaneli loves Starburst candies, but wants to keep her snacks to 100 calories. If the candies have 160 calories for 8 pieces, how many pieces can she have in her snack?

ჲრკპთირვ ჲრდჲგჲპა.

$5\ \text{pieces}$

Josiah went to Mexico for spring break and changed $325 dollars into Mexican pesos. At that time, the exchange rate had $1 US is equal to 12.54 Mexican pesos. How many Mexican pesos did he get for his trip?

ჲრკპთირვ ჲრდჲგჲპა.

What are you asked to find?		How many Mexican pesos did Josiah get?
Assign a variable.		Let $p=\text{the number of Mexican pesos.}$
Write a sentence that gives the information to find it.		If $1 US is equal to 12.54 Mexican pesos, then $325 is how many pesos?
Translate into a proportion–be careful of the units.
$\frac{\text{\$}}{\text{pesos}}=\frac{\text{\$}}{\text{pesos}}$
Multiply both sides by the LCD, $12.54p$.
Remove common factors on each side.
Simplify.
Check.
Is the answer reasonable?
Yes, $100 would be 1,254 pesos. $325 is a little more than 3 times this amount, so our answer of 4075.5 pesos makes sense.

Write a complete sentence.		Josiah got 4075.5 pesos for his spring break trip.

Yurianna is going to Europe and wants to change $800 dollars into Euros. At the current exchange rate, $1 US is equal to 0.738 Euro. How many Euros will she have for her trip?

ჲრკპთირვ ჲრდჲგჲპა.

$590.4\ \text{Euros}$
Corey and Nicole are traveling to Japan and need to exchange $600 into Japanese yen. If each dollar is 94.1 yen, how many yen will they get?

ჲრკპთირვ ჲრდჲგჲპა.

$56,460\ \text{yen}$

$\text{\Delta }ABC$ is similar to $\text{\Delta }XYZ$. The lengths of two sides of each triangle are given. Find the lengths of the third sides.

ჲრკპთირვ ჲრდჲგჲპა.

Step 1. Read the problem. Draw the figure and label it with the given information.	Figure is given.
Step 2. Identify what we are looking for.	the length of the sides of similar triangles
Step 3. Name the variables.	Let $\ a=$ length of the third side of $\text{\Delta }ABC.$ $y=$ length of the third side of $\text{\Delta }XYZ$
Step 4. Translate.	Since the triangles are similar, the corresponding sides are proportional.
We need to write an equation that compares the side we are looking for to a known ratio. Since the side AB = 4 corresponds to the side XY = 3 we know $\frac{AB}{XY}=\frac{4}{3}$. So we write equations with $\frac{AB}{XY}$ to find the sides we are looking for. Be careful to match up corresponding sides correctly.	$\frac{AB}{XY}=\frac{BC}{YZ}=\frac{AC}{XZ}.$
Substitute.	$\$$\$
Step 5. Solve the equation.	$\$$\$
	$\$$\$
Step 6. Check. $\begin{array}{llllllllllllllllll} \\ \\ \begin{array}{lll}\frac{4}{3} & \overset{?}{=} & \frac{6}{4.5} \\ 4(4.5) & \overset{?}{=} & 6(3) \\ 18 & = & 18✓\end{array} & & & & & \begin{array}{lll}\frac{4}{3} & \overset{?}{=} & \frac{3.2}{2.4} \\ 4(2.4) & \overset{?}{=} & 3.2(3) \\ 9.6 & = & 9.6✓\end{array}\end{array}$
Step 7. Answer the question.	The third side of $\text{\Delta }ABC$ is 6 and the third side of $\text{\Delta }XYZ$ is 2.4.

$\text{\Delta }ABC$ is similar to $\text{\Delta }XYZ$. The lengths of two sides of each triangle are given in the figure.
Find the length of side $a$.

ჲრკპთირვ ჲრდჲგჲპა.

8
$\text{\Delta }ABC$ is similar to $\text{\Delta }XYZ$. The lengths of two sides of each triangle are given in the figure.
Find the length of side $y$.

ჲრკპთირვ ჲრდჲგჲპა.

22.5

On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. If the actual distance from Los Angeles to Las Vegas is 270 miles find the distance from Los Angeles to San Francisco.

ჲრკპთირვ ჲრდჲგჲპა.

Read the problem. Draw the figures and label with the given information.	The figures are shown above.
Identify what we are looking for.	The actual distance from Los Angeles to San Francisco.
Name the variables.	Let $x=$ distance from Los Angeles to San Francisco.
Translate into an equation. Since the triangles are similar, the corresponding sides are proportional. We'll make the numerators "miles" and the denominators "inches."
Solve the equation.

Check.
On the map, the distance from Los Angeles to San Francisco is more than the distance from Los Angeles to Las Vegas. Since 351 is more than 270 the answer makes sense.

Answer the question.	The distance from Los Angeles to San Francisco is 351 miles.

On the map, Seattle, Portland, and Boise form a triangle whose sides are shown in the figure below. If the actual distance from Seattle to Boise is 400 miles, find the distance from Seattle to Portland.

ჲრკპთირვ ჲრდჲგჲპა.

150 miles
Using the map above, find the distance from Portland to Boise.

ჲრკპთირვ ჲრდჲგჲპა.

350 miles

Tyler is 6 feet tall. Late one afternoon, his shadow was 8 feet long. At the same time, the shadow of a tree was 24 feet long. Find the height of the tree.

ჲრკპთირვ ჲრდჲგჲპა.

Read the problem and draw a figure.
We are looking for h, the height of the tree.
We will use similar triangles to write an equation.
The small triangle is similar to the large triangle.
Solve the proportion.
Simplify.
Check.
Tyler's height is less than his shadow's length so it makes sense that the tree's height is less than the length of its shadow.

A telephone pole casts a shadow that is 50 feet long. Nearby, an 8 foot tall traffic sign casts a shadow that is 10 feet long. How tall is the telephone pole?

ჲრკპთირვ ჲრდჲგჲპა.

40 feet
A pine tree casts a shadow of 80 feet next to a 30-foot tall building which casts a 40 feet shadow. How tall is the pine tree?

ჲრკპთირვ ჲრდჲგჲპა.

60 feet
$\frac{x}{56}=\frac{7}{8}$

ჲრკპთირვ ჲრდჲგჲპა.

$49$
$\frac{n}{91}=\frac{8}{13}$
$\frac{49}{63}=\frac{z}{9}$

ჲრკპთირვ ჲრდჲგჲპა.

$7$
$\frac{56}{72}=\frac{y}{9}$
$\frac{5}{a}=\frac{65}{117}$

ჲრკპთირვ ჲრდჲგჲპა.

$9$
$\frac{4}{b}=\frac{64}{144}$
$\frac{98}{154}=\frac{-7}{p}$

ჲრკპთირვ ჲრდჲგჲპა.

$-11$
$\frac{72}{156}=\frac{-6}{q}$

Symbols used here

f'(x),\ \frac{dy}{dx}

derivative
Instantaneous rate of change; slope of the graph.

\neq

not equal
The two sides are different.

\pm

plus or minus
Both signs at once: x = 3 ± 2 means 5 and 1.

\leq,\ \geq

less/greater than or equal
Inequalities that allow equality; < and > exclude it.

\sqrt{x},\ \sqrt[n]{x}

square root, n-th root
The non-negative number whose square (n-th power) is x.

|x|

absolute value / modulus
Distance from zero: |−3| = 3. For a complex number, distance from the origin.

imaginary unit
i² = −1.

\log_b x,\ \ln x

logarithm, natural log
The exponent b must be raised to for x; ln uses base e.

\mathbb{N},\ \mathbb{Z},\ \mathbb{Q},\ \mathbb{R},\ \mathbb{C}

number sets
Naturals, integers, rationals, reals, complex numbers.

How to: Solve Proportion and Similar Figure Applications

Solve proportions
Solve similar figure applications
If

Questions people ask

What does it mean to solve an equation?

To find every value of the unknown that makes both sides equal. Each step is an operation applied to both sides that keeps the solution set the same, until the unknown stands alone.

Why do I sometimes get two answers?

A quadratic can cross the axis twice, so it can have two solutions. A degree-n polynomial has up to n. The graph shows where each one comes from.

How do I know whether to factor or use the quadratic formula?

Try factoring for a few seconds: look for two numbers that multiply to a·c and add to b. If nothing obvious appears, the discriminant b² − 4ac tells you how many real roots there are, and the formula finds them without guessing.

ჲოთრაი ჟამ.

Parts of this page are adapted from OpenStax Elementary Algebra 2e (CC BY-NC-SA 4.0). Condensed and re-explained here; errors are ours.

მეტი Algebra

Linear equations Quadratic equations Systems of equations Inequalities Factoring Expanding Simplifying expressions Functions and graphs Exponential and logarithmic equations Polynomial equations Absolute value


To isolate \(x\), multiply both sides by the LCD, 63.
Simplify.
Divide the common factors.
Check. To check our answer, we substitute into the original proportion.


Show common factors.
Simplify.


Multiply both sides by the LCD, 18.
Simplify.
Distribute.
Solve for \(p\).
Check.


Simplify.
Divide.

Identify what we are asked to find, and choose a variable to represent it.		How many ml of acetaminophen will the doctor prescribe?
		Let \(a=\text{ml}\) of acetaminophen.
Write a sentence that gives the information to find it.		If 5 ml is prescribed for every 25 pounds, how much will be prescribed for 80 pounds?
Translate into a proportion–be careful of the units. \(\frac{\text{ml}}{\text{pounds}}=\frac{\text{ml}}{\text{pounds}}\)
Multiply both sides by the LCD, 400.
Remove common factors on each side.
Simplify, but don't multiply on the left. Notice what the next step will be.
Solve for \(a\).
Check.
Is the answer reasonable?
Yes, since 80 is about 3 times 25, the medicine should be about 3 times 5. So 16 ml makes sense.

Write a complete sentence.		The pediatrician would prescribe 16 ml of acetaminophen to Zoe.

Identify what we are asked to find, and choose a variable to represent it.		How many calories are in a 24 ounce iced caramel macchiato?
		Let \(c=\text{calories}\) in 24 ounces.
Write a sentence that gives the information to find it.		If there are 230 calories in 16 ounces, then how many calories are in 24 ounces?
Translate into a proportion–be careful of the units. \(\frac{\text{calories}}{\text{ounce}}=\frac{\text{calories}}{\text{ounce}}\)
Multiply both sides by the LCD, 48.
Remove common factors on each side.
Simplify.
Solve for \(c\).

Check.
Is the answer reasonable?
Yes, 345 calories for 24 ounces is more than 290 calories for 16 ounces, but not too much more.

Write a complete sentence.		There are 345 calories in a 24-ounce iced caramel macchiato.