homework 1 3 applications of ratios

Ratio Worksheets

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Ratios in the Classroom

I always loved teaching ratios.  Students often did well with the concept, I was able to incorporate fun activities, and we quickly moved on to proportions .  With the changes in the Common Core Standards, students are expected to have a much deeper understanding of ratios.

6.RP.1 Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities.  For example, “The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak.” “For every vote candidate A received, candidate C received nearly three votes.” 6.RP.3 Use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations.

This can be tricky for us as teachers, as it is more complex than what we learned or have experience teaching.  I think we are all familiar with 6.RP.1, using ratio language.  We often practice writing it three different ways and comparing part-to-part relationships vs. part-to-whole relationships.

When I review the standards, there are a few things that really pop out, specifically 6.RP.3, “use ratio reasoning to solve…”, and the various models noted.   I am guilty of often making the leap from ratios to proportions very quickly, while possibly leaving some gaps as to how they are related.  I think this is where the specific models come into play.

Ratio Models

The standards specifically mention three models before moving on to equations.  I am going to give it my best effort in explaining how they work and what things to make note of when teaching them.  As a side note, I would encourage you to use the different models throughout the unit while working problems.  It is easy to get in the habit of teaching all three and then only use one for the remainder of the unit.  I noticed this during my first year of teaching Algebra in our polynomials unit.  I preferred to double distribute; the teacher across the hall preferred the box method.  During our common tutorial time, any student in my class double distributed, while all of hers used the box method.  It really goes to show that exposing students to the various models regularly helps them be more well-rounded and in the case of these standards, use all three models fluidly.

Tables of Equivalent Ratios

I love this model, mostly because it is so applicable for future math courses, including Algebra.  It is easy to draw and takes up less space than the others.

Additionally, there are a few ratio problems that involve changing the ratio.  In these cases, it is helpful to add the third column “total” to the ratio table, as well as recognize that there are two different ratios, therefore two tables. The example above is perfect for a transition to proportions when the time is ready.

Double Number Lines

Tape Diagrams

Lastly, tape diagrams are a more concrete version of a table or double number line.  These might be perfect with students who are struggling to move toward a more abstract understanding and can even be represented with hands-on fraction bars or Cuisinart rods.  Be on the lookout for students who draw different sized boxes and thus change the ratio without realizing it.

Common Misconceptions

There are quite a few common misconceptions that students might have.  It is helpful to not only address these while teaching but even possibly have students analyze work to see if the misconceptions exist.  We often graded famous people’s homework assignments in class to practice error analysis, and it’s perfect for incorporating those mathematical practices. It’s a win-win!

• Mixing up the ratio because they didn’t read the problem
• Not recognizing that the comparison involves the total or difference
• Using addition to describe the relationship

Ideas for Struggling Students

• Sort part-to-part and part-to-whole relationships without working the problems.
• Match situations to the appropriate model.
• Match different models together.
• Practice finding equivalent ratios.
• Given a model, students write a problem.

Ratios provide a foundation to proportional relationships and reasoning.  Be sure to spend the necessary time to make sure your students are confident in their reasoning skills and can apply the models appropriately.

Be sure to check out these different concepts and activities that are included in my Ratios Unit and Ratios Activity Bundle .

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Unit 4: Proportional relationships and percentages

Lesson 2: ratios and rates with fractions.

• No videos or articles available in this lesson
• Rates with fractions Get 3 of 4 questions to level up!

Lesson 3: Revisiting proportional relationships

• Proportion word problems Get 3 of 4 questions to level up!

Lesson 5: Say it with decimals

• Fraction to decimal: 11/25 (Opens a modal)
• Worked example: Converting a fraction (7/8) to a decimal (Opens a modal)
• Converting fractions to decimals Get 3 of 4 questions to level up!

Lesson 7: One hundred percent

• Rational number word problem: ice (Opens a modal)
• Percent problems Get 3 of 4 questions to level up!

Lesson 8: Percent increase and decrease with equations

• Equivalent expressions with percent problems Get 3 of 4 questions to level up!

Lesson 10: Tax and tip

• Interpreting linear expressions: diamonds (Opens a modal)
• Tax and tip word problems Get 3 of 4 questions to level up!

Lesson 11: Percentage contexts

• Percent word problem: guavas (Opens a modal)
• Percent word problems: tax and discount (Opens a modal)
• Discount, markup, and commission word problems Get 3 of 4 questions to level up!

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Lesson 3 Ratio Applications

Lesson 3 Ratio Applications - Displaying top 8 worksheets found for this concept.

Some of the worksheets for this concept are Learning to think mathematically with the ratio table, Chapter 6 ratio and proportion, Eureka lessons for 7th grade unit three ratios, Fibonacci is all around, Ratios rates unit rates, Right triangle applications, Solving ratio and percent problems using proportional, A story of ratios.

Found worksheet you are looking for? To download/print, click on pop-out icon or print icon to worksheet to print or download. Worksheet will open in a new window. You can & download or print using the browser document reader options.

1. Learning to Think Mathematically with the Ratio Table

2. chapter 6 ratio and proportion, 3. eureka lessons for 7th grade unit three ~ ratios ..., 4. fibonacci is all around, 5. ratios, rates & unit rates, 6. right triangle applications, 7. solving ratio and percent problems using proportional ..., 8. a story of ratios.

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3.1.6: Applications of Percents

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• Denny Burzynski & Wade Ellis, Jr.
• College of Southern Nevada via OpenStax CNX

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Learning Objectives

• be able to distinguish between base, percent, and percentage
• be able to find the percentage, the percent, and the base

Base, Percent, and Percentage

There are three basic types of percent problems. Each type involves a base, a percent, and a percentage, and when they are translated from words to mathemati­cal symbols each becomes a multiplication statement . Examples of these types of problems are the following:

• What number is 30% of 50? (Missing product statement.)
• 15 is what percent of 50? (Missing factor statement.)
• 15 is 30% of what number? (Missing factor statement.)

In problem 1, the product is missing. To solve the problem, we represent the missing product with \(P\).

\(P = 30\% \cdot 50\)

Definition: Percentage

The missing product \(P\) is called the percentage . Percentage means part , or por­tion . In \(P = 30\% \cdot 50\). \(P\) represents a particular part of 50.

In problem 2, one of the factors is missing. Here we represent the missing factor with \(Q\).

\(15 = Q \cdot 50\)

Percent The missing factor is the percent . Percent, we know, means per 100, or part of 100. In \(15 = Q \cdot 50\). \(Q\) indicates what part of 50 is being taken or considered. Specifi­cally, \(15 = Q \cdot 50\) means that if 50 was to be divided into 100 equal parts, then \(Q\) indicates 15 are being considered.

In problem 3, one of the factors is missing. Represent the missing factor with \(B\).

\(15 = 30\% \cdot B\)

Base The missing factor is the base . Some meanings of base are a source of supply , or a starting place . In \(15 = 30\% \cdot B\), \(B\) indicates the amount of supply. Specifically, \(15 = 30\% \cdot B\) indicates that 15 represents 30% of the total supply.

Each of these three types of problems is of the form

\(\text{(percentage)} = \text{(percent)} \cdot \text{(base)}\)

We can determine any one of the three values given the other two using the methods discussed in [link] .

Finding the Percentage

Sample Set A

\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {30\%} & {\text{of}} & {50?} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {30\%} & {\cdot} & {50} & {\text{Convert 30% to a decimal.}} \\ {P} & {=} & {.30} & {\cdot} & {50} & {\text{Multiply.}} \\ {P} & {=} & {15} & {} & {} & {} \end{array}\)

Thus, 15 is 30% of 50.

\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {36\%} & {\text{of}} & {95?} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {36\%} & {\cdot} & {95} & {\text{Convert 36% to a decimal.}} \\ {P} & {=} & {.36} & {\cdot} & {95} & {\text{Multiply.}} \\ {P} & {=} & {34.2} & {} & {} & {} \end{array}\)

Thus, 34.2 is 36% of 95.

A salesperson, who gets a commission of 12% of each sale she makes, makes a sale of $8,400.00. How much is her commission?

We need to determine what part of $8,400.00 is to be taken. What part indicates percentage .

\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {12\%} & {\text{of}} & {8,400.00?} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {12\%} & {\cdot} & {8,400.00} & {\text{Convert to decimals.}} \\ {P} & {=} & {.12} & {\cdot} & {8,400.00} & {\text{Multiply.}} \\ {P} & {=} & {1008.00} & {} & {} & {} \end{array}\)

Thus, the salesperson's commission is $1,008.00.

A girl, by practicing typing on her home computer, has been able to increase her typing speed by 110%. If she originally typed 16 words per minute, by how many words per minute was she able to increase her speed?

We need to determine what part of 16 has been taken. What part indicates percentage .

\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {110\%} & {\text{of}} & {16?} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {110\%} & {\cdot} & {16} & {\text{Convert to decimals.}} \\ {P} & {=} & {1.10} & {\cdot} & {16} & {\text{Multiply.}} \\ {P} & {=} & {17.6} & {} & {} & {} \end{array}\)

Thus, the girl has increased her typing speed by 17.6 words per minute. Her new speed is \(16 + 17.6 = 33.6\) words per minute.

A student who makes $125 a month working part-time receives a 4% salary raise. What is the student's new monthly salary?

With a 4% raise, this student will make 100% of the original salary + 4% of the original salary. This means the new salary will be 104% of the original salary. We need to determine what part of $125 is to be taken. What part indicates percentage .

\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {104\%} & {\text{of}} & {125} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {104\%} & {\cdot} & {125} & {\text{Convert to decimals.}} \\ {P} & {=} & {1.04} & {\cdot} & {125} & {\text{Multiply.}} \\ {P} & {=} & {130} & {} & {} & {} \end{array}\)

Thus, this student's new monthly salary is $130.

An article of clothing is on sale at 15% off the marked price. If the marked price is $24.95, what is the sale price?

Since the item is discounted 15%, the new price will be \(100\% - 15\% = 85\%\) of the marked price. We need to determine what part of 24.95 is to be taken. What part indicates percentage .

\(\begin{array} {cccccl} {\text{What number}} & {\text{is}} & {85\%} & {\text{of}} & {$24.95} & {\text{Missing product statement.}} \\ {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {P} & {=} & {85\%} & {\cdot} & {24.95} & {\text{Convert to decimals.}} \\ {P} & {=} & {.85} & {\cdot} & {24.95} & {\text{Multiply.}} \\ {P} & {=} & {21.2075} & {} & {} & {\text{Since this number represents money,}} \\ {} & {} & {} & {} & {} & {\text{we'll round to 2 decimal places}} \\ {P} & {=} & {21.21} & {} & {} & {} \end{array}\)

Thus, the sale price of the item is $21.21.

Practice Set A

What number is 42% of 85?

A sales person makes a commission of 16% on each sale he makes. How much is his commission if he makes a sale of $8,500?

An assembly line worker can assemble 14 parts of a product in one hour. If he can increase his assembly speed by 35%, by how many parts per hour would he increase his assembly of products?

A computer scientist in the Silicon Valley makes $42,000 annually. What would this scientist's new annual salary be if she were to receive an 8% raise?

Finding the Percent

Sample Set B

\(\begin{array} {cccccl} {15} & {\text{is}} & {\text{What number}} & {\text{of}} & {50?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {15} & {=} & {Q} & {\cdot} & {50} & {} \end{array}\)

Recall that (missing factor) = (product) \(\div\) (known factor).

\(\begin{array} {rccl} {Q} & = & {15 \div 50} & {\text{Divide.}} \\ {Q} & = & {0.3} & {\text{Convert to a percent}} \\ {Q} & = & {30\%} & {} \end{array}\)

\(\begin{array} {cccccl} {4.32} & {\text{is}} & {\text{What percent}} & {\text{of}} & {72?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {4.32} & {=} & {Q} & {\cdot} & {72} & {} \end{array}\)

\(\begin{array} {rccl} {Q} & = & {4.32 \div 72} & {\text{Divide.}} \\ {Q} & = & {0.06} & {\text{Convert to a percent}} \\ {Q} & = & {6\%} & {} \end{array}\)

Thus, 4.32 is 6% of 72.

On a 160 question exam, a student got 125 correct answers. What percent is this? Round the result to two decimal places.

We need to determine the percent.

\(\begin{array} {cccccl} {125} & {\text{is}} & {\text{What percent}} & {\text{of}} & {160?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {125} & {=} & {Q} & {\cdot} & {160} & {} \end{array}\)

\(\begin{array} {rccl} {Q} & = & {125 \div 160} & {\text{Divide.}} \\ {Q} & = & {0.78125} & {\text{Round to two decimal places}} \\ {Q} & = & {.78} & {} \end{array}\)

Thus, this student received a 78% on the exam.

A bottle contains 80 milliliters of hydrochloric acid (HCl) and 30 milliliters of water. What percent of HCl does the bottle contain? Round the result to two decimal places.

We need to determine the percent. The total amount of liquid in the bottle is

\(\text{80 milliliters + 30 milliliters = 110 milliliters}\)

\(\begin{array} {cccccl} {80} & {\text{is}} & {\text{What percent}} & {\text{of}} & {110?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {80} & {=} & {Q} & {\cdot} & {110} & {} \end{array}\)

\(\begin{array} {rccl} {Q} & = & {80 \div 110} & {\text{Divide.}} \\ {Q} & = & {0.727272...} & {\text{Round to two decimal places}} \\ {Q} & \approx & {73\%} & {\text{The symbol "} \approx \text{" is read as "approximately."}} \end{array}\)

Thus, this bottle contains approximately 73% HCl.

Five years ago a woman had an annual income of $19,200. She presently earns $42,000 annually. By what percent has her salary increased? Round the result to two decimal places.

\(\begin{array} {cccccl} {42,000} & {\text{is}} & {\text{What percent}} & {\text{of}} & {19,200?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {42,000} & {=} & {Q} & {\cdot} & {19,200} & {} \end{array}\)

\(\begin{array} {rccl} {Q} & = & {42,000 \div 19,200} & {\text{Divide.}} \\ {Q} & = & {2.1875} & {\text{Round to two decimal places}} \\ {Q} & = & {2.19} & {\text{Convert to a percent.}} \\ {Q} & = & {219\%} & {\text{Convert to a percent.}} \end{array}\)

Thus, this woman's annual salary has increased 219%.

Practice Set B

99.13 is what percent of 431?

On an 80 question exam, a student got 72 correct answers. What percent did the student get on the exam?

A bottle contains 45 milliliters of sugar and 67 milliliters of water. What fraction of sugar does the bottle contain? Round the result to two decimal places (then express as a percent).

Finding the Base

Sample Set C

\(\begin{array} {cccccl} {15} & {\text{is}} & {30\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {15} & {=} & {30\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {15} & {=} & {.30} & {\cdot} & {B} & {\text{[\text{(missing factor)} = \text{(product)} \div \text{(known factor)}]}} \end{array}\)

\(\begin{array} {rcl} {B} & = & {15 \div .30} \\ {B} & = & {50} \end{array}\)

\(\begin{array} {cccccl} {56.43} & {\text{is}} & {33\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {56.43} & {=} & {33\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {56.43} & {=} & {.33} & {\cdot} & {B} & {\text{Divide.}} \end{array}\)

\(\begin{array} {rcl} {B} & = & {56.43 \div .33} \\ {B} & = & {171} \end{array}\)

Thus, 56.43 is 33% of 171.

Fifteen milliliters of water represents 2% of a hydrochloric acid (HCl) solution. How many milliliters of solution are there?

We need to determine the total supply. The word supply indicates base .

\(\begin{array} {cccccl} {15} & {\text{is}} & {2\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {[\text{(product)} = \text{(factor)} \cdot \text{(factor)}]} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {15} & {=} & {2\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {15} & {=} & {.02} & {\cdot} & {B} & {\text{Divide.}} \end{array}\)

\(\begin{array} {rcl} {B} & = & {15 \div .02} \\ {B} & = & {750} \end{array}\)

Thus, there are 750 milliliters of solution in the bottle.

In a particular city, a sales tax of \(6 \dfrac{1}{2}\) % is charged on items purchased in local stores. If the tax on an item is $2.99, what is the price of the item?

We need to determine the price of the item. We can think of price as the starting place . Starting place indicates base . We need to determine the base.

\(\begin{array} {cccccl} {2.99} & {\text{is}} & {6 \dfrac{1}{2}\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {2.99} & {=} & {6 \dfrac{1}{2}\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {2.99} & {=} & {6.5\%} & {\cdot} & {B} & {} \\ {2.99} & {=} & {0.065} & {\cdot} & {B} & {[\text{(missing factor)} = \text{(product)} \div \text{(known factor)}]} \end{array}\)

\(\begin{array} {rcll} {B} & = & {2.99 \div .065} & {\text{Divide.}} \\ {B} & = & {46} & {} \end{array}\)

Thus, the price of the item is $46.00.

A clothing item is priced at $20.40. This marked price includes a 15% discount. What is the original price?

We need to determine the original price. We can think of the original price as the starting place . Starting place indicates base . We need to determine the base. The new price, $20.40, represents \(100\% - 15\% = 85\%\) of the original price.

\(\begin{array} {cccccl} {20.40} & {\text{is}} & {85\%} & {\text{of}} & {\text{What number}?} & {\text{Missing factor statement.}} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\underbrace{\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }_{}}} & {} \\ {\text{(percentage)}} & {=} & {\text{(percent)}} & {\cdot} & {\text{(base)}} & {} \\ {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {^{\downarrow}} & {} \\ {20.40} & {=} & {85\%} & {\cdot} & {B} & {\text{Convert to decimals.}} \\ {20.40} & {=} & {0.85} & {\cdot} & {B} & {[\text{(missing factor)} = \text{(product)} \div \text{(known factor)}]} \end{array}\)

\(\begin{array} {rcll} {B} & = & {20.40 \div .85} & {\text{Divide.}} \\ {B} & = & {24} & {} \end{array}\)

Thus, the original price of the item is $24.00.

Practice Set C

1.98 is 2% of what number?

3.3 milliliters of HCl represents 25% of an HCl solution. How many milliliters of solution are there?

A salesman, who makes a commission of \(18 \dfrac{1}{4}\)% on each sale, makes a commission of $152.39 on a particular sale. Rounded to the nearest dollar, what is the amount of the sale?

At "super-long play," \(2\dfrac{1}{2}\) hours of play of a video cassette recorder represents 31.25% of the total playing time. What is the total playing time?

For the following 25 problems, find each indicated quantity.

Exercise \(\PageIndex{1}\)

What is 21% of 104?

Exercise \(\PageIndex{2}\)

What is 8% of 36?

Exercise \(\PageIndex{3}\)

What is 98% of 545?

Exercise \(\PageIndex{4}\)

What is 143% of 33?

Exercise \(\PageIndex{5}\)

What is \(10 \dfrac{1}{2}\)% of 20?

Exercise \(\PageIndex{6}\)

3.25 is what percent of 88?

Exercise \(\PageIndex{7}\)

22.44 is what percent of 44?

Exercise \(\PageIndex{8}\)

0.0036 is what percent of 0.03?

Exercise \(\PageIndex{9}\)

31.2 is what percent of 26?

Exercise \(\PageIndex{10}\)

266.4 is what percent of 74?

Exercise \(\PageIndex{11}\)

0.0101 is what percent of 0.0505?

Exercise \(\PageIndex{12}\)

2.4 is 24% of what number?

Exercise \(\PageIndex{13}\)

24.19 is 41% of what number?

Exercise \(\PageIndex{14}\)

61.12 is 16% of what number?

Exercise \(\PageIndex{15}\)

82.81 is 91% of what number?

Exercise \(\PageIndex{16}\)

115.5 is 20% of what number?

Exercise \(\PageIndex{17}\)

43.92 is 480% of what number?

Exercise \(\PageIndex{18}\)

What is 85% of 62?

Exercise \(\PageIndex{19}\)

29.14 is what percent of 5.13?

Exercise \(\PageIndex{20}\)

0.6156 is what percent of 5.13?

Exercise \(\PageIndex{21}\)

What is 0.41% of 291.1?

Exercise \(\PageIndex{22}\)

26.136 is 121% of what number?

Exercise \(\PageIndex{23}\)

1,937.5 is what percent of 775?

Exercise \(\PageIndex{24}\)

1 is what percent of 2,000?

Exercise \(\PageIndex{25}\)

0 is what percent of 59?

Exercise \(\PageIndex{26}\)

An item of clothing is on sale for 10% off the marked price. If the marked price is $14.95, what is the sale price? (Round to two decimal places.)

Exercise \(\PageIndex{27}\)

A grocery clerk, who makes $365 per month, re­ceives a 7% raise. How much is her new monthly salary?

Exercise \(\PageIndex{28}\)

An item of clothing which originally sells for $55.00 is marked down to $46.75. What percent has it been marked down?

Exercise \(\PageIndex{29}\)

On a 25 question exam, a student gets 21 correct. What percent is this?

Exercise \(\PageIndex{30}\)

On a 45 question exam, a student gets 40%. How many questions did this student get correct?

Exercise \(\PageIndex{31}\)

A vitamin tablet, which weighs 250 milligrams, contains 35 milligrams of vitamin C. What per­cent of the weight of this tablet is vitamin C?

Exercise \(\PageIndex{32}\)

Five years ago a secretary made $11,200 an­nually. The secretary now makes $17,920 an­nually. By what percent has this secretary's sal­ary been increased?

Exercise \(\PageIndex{33}\)

A baseball team wins \(48 \dfrac{3}{4}\)% of all their games. If they won 78 games, how many games did they play?

Exercise \(\PageIndex{34}\)

A typist was able to increase his speed by 120% to 42 words per minute. What was his original typ­ing speed?

Exercise \(\PageIndex{35}\)

A salesperson makes a commission of 12% on the total amount of each sale. If, in one month, she makes a total of $8,520 in sales, how much has she made in commission?

Exercise \(\PageIndex{36}\)

A salesperson receives a salary of $850 per month plus a commission of \(8\dfrac{1}{2}\) % of her sales. If, in a particular month, she sells $22,800 worth of merchandise, what will be her monthly earn­ings?

Exercise \(\PageIndex{37}\)

A man borrows $1150.00 from a loan company. If he makes 12 equal monthly payments of $130.60, what percent of the loan is he paying in interest?

Exercise \(\PageIndex{38}\)

The distance from the sun to the earth is approx­imately 93,000,000 miles. The distance from the sun to Pluto is approximately 860.2% of the dis­tance from the sun to the Earth. Approximately, how many miles is Pluto from the sun?

Exercise \(\PageIndex{39}\)

The number of people on food stamps in Maine in 1975 was 151,000. By 1980, the number had decreased to 59,200. By what percent did the number of people on food stamps decrease? (Round the result to the nearest percent.)

Exercise \(\PageIndex{40}\)

In Nebraska, in 1960, there were 734,000 motor-vehicle registrations. By 1979, the total had in­creased by about 165.6%. About how many motor-vehicle registrations were there in Ne­braska in 1979?

Exercise \(\PageIndex{41}\)

From 1973 to 1979, in the United States, there was an increase of 166.6% of Ph.D. social scien­tists to 52,000. How many were there in 1973?

Exercise \(\PageIndex{42}\)

In 1950, in the United States, there were 1,894 daily newspapers. That number decreased to 1,747 by 1981. What percent did the number of daily newspapers decrease?

Exercise \(\PageIndex{43}\)

A particular alloy is 27% copper. How many pounds of copper are there in 55 pounds of the alloy?

Exercise \(\PageIndex{44}\)

A bottle containing a solution of hydrochloric acid (HCl) is marked 15% (meaning that 15% of the HCl solution is acid). If a bottle contains 65 milliliters of solution, how many milliliters of water does it contain?

Exercise \(\PageIndex{45}\)

A bottle containing a solution of HCl is marked 45%. A test shows that 36 of the 80 milliliters contained in the bottle are hydrochloric acid. Is the bottle marked correctly? If not, how should it be remarked?

Marked correctly

Exercises For Review

Exercise \(\PageIndex{46}\)

Use the numbers 4 and 7 to illustrate the commutative property of multiplication.

Exercise \(\PageIndex{47}\)

Convert \(\dfrac{14}{5}\) to a mixed number.

\(2\dfrac{4}{5}\)

Exercise \(\PageIndex{48}\)

Arrange the numbers \(\dfrac{7}{12}\), \(\dfrac{5}{9}\) and \(\dfrac{4}{7}\) in increasing order.

Exercise \(\PageIndex{49}\)

Convert 4.006 to a mixed number.

\(4 \dfrac{3}{500}\)

Exercise \(\PageIndex{50}\)

Convert \(\dfrac{7}{8}\)% to a fraction.