Algebra · Chapter 11

Ratios and Percentages

Split totals into equal parts, chain percents the right way, and add up work rates to see how fast a team finishes.

What's this chapter about?

11.1 Ratios Fundamentals
11.2 Rate and Work
11.3 Practice Problems

You'll need

Nothing! This is where it all starts. If you can multiply, you're ready.

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11.10%

Ratios Fundamentals

In this section you will be able to split a total into equal parts using a ratio, and take a percent of a percent.

A class has boys and girls in the ratio 3 : 5, and 40 kids in all. How many girls are there? Draw 8 equal boxes and you are almost done!

A ratio like 3 : 5 says: cut the whole into $3 + 5 = 8$ equal parts. Give 3 parts to one group and 5 to the other. Find what one part is worth, and every question becomes multiplication.

A percent is a ratio out of 100. To take a percent of a percent, multiply: 40% of the marbles are large, and 10% of those are yellow, so $0.4 \times 0.1 = 0.04$ , or 4%, are large and yellow.

Look:a percent change always acts on the current amount, not the start. That is why +20% then −20% leaves you at 96%, not 100%.

Exploreplay first, formulas later

one part =1

Goal: total = 40

Boys3 parts = 3

Girls5 parts = 5

colour	Boys	Girls	all
boys	3 parts = 3	0 parts = 0	3 parts = 3
girls	0 parts = 0	5 parts = 5	5 parts = 5

8 parts × 1 = 8

Try this: The ratio is 3 : 5 and the goal is a total of 40. Raise "one part" until the goal turns green. What is one part worth?
Try this: How many more girls than boys are there? Count the extra boxes first, then multiply by one part.
Try this: Press + Add a bar and make the ratio 2 : 3 : 5. If the total were 50, what would one part be? Check it with the stepper.

Watch itpredict, then reveal

Example 11.1

MATHCOUNTS

The Ten Finger calculator company periodically checks random calculators before shipping crates out to customers. On Wednesday, 12 calculators from each of 64 crates of 144 calculators were tested. Two of the tested calculators were found to be defective. Based on this rate of defect, how many total calculators are expected to be defective?

colour	all
tested group	1 parts = ?
other groups	11 parts = ?

1
What fraction of the total calculators were tested?

Predict first:
Your prediction

Example 11.2

Omega Learn

Sohil has a jar with many marbles. 20% of them are small, 40% are medium, and 40% are large. For each size, 30% of the marbles are red, 20% blue, 40% green, and 10% yellow. Half of large yellow and small green marbles have a special design on them. What percentage of the total marbles have a special design?

1
What percent of marbles are large and yellow?

Predict first:
Your prediction

Example 11.3

AMC 10

Two jars each contain the same number of marbles, and every marble is either blue or green. In Jar 1 the ratio of blue to green marbles is $9:1$ , and the ratio of blue to green marbles in Jar 2 is $8:1$ . There are 95 green marbles in all. How many more blue marbles are in Jar 1 than in Jar 2?

one part =1

Jar 110 parts = ?

Jar 29 parts = ?

colour	Jar 1	Jar 2	all
blue	9 parts = ?	8 parts = ?	17 parts = ?
green	1 parts = ?	1 parts = ?	2 parts = ?

19 parts × ? = ?

1
Look at the two tapes. Jar 1 has $9 + 1$ parts and Jar 2 has $8 + 1$ parts. Can both tapes use the same part size?

Predict first:

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Example 11.4

Omega Learn

There are 994 students in the Omega Middle School. $\frac{2}{7}$ of them are in 6th grade, $\frac{2}{7}$ of them are in 7th grade, and $\frac{3}{7}$ of them are in 8th grade. $\frac{1}{4}$ of the 6th graders are interested in math, $\frac{3}{4}$ of the 7th graders are interested in math, and $\frac{1}{2}$ of the 8th graders are interested in math. Of the students who are interested in math, $\frac{5}{7}$ will take the AMC 8 this year. How many students will take the AMC 8 from Omega Middle School?

1
How many students are in each grade?

Predict first:
Your prediction: 6thYour prediction: 7thYour prediction: 8th

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Example 11.5

AMC 8

Steph scored 15 baskets out of 20 attempts in the first half of a game, and 10 baskets out of 10 attempts in the second half. Candace took 12 attempts in the first half and 18 attempts in the second. In each half, Steph scored a higher percentage of baskets than Candace. Surprisingly they ended with the same overall percentage of baskets scored. How many more baskets did Candace score in the second half than in the first?

	First Half	Second Half
Steph	$\frac{15}{20}$	$\frac{10}{10}$
Candace	$\frac{\square}{12}$	$\frac{\square}{18}$

1
How many baskets did Steph score in all, out of how many attempts?

Predict first:
Your prediction

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Problem 11.3.1

AMC 8

If the degree measures of the angles of a triangle are in the ratio $3:3:4$ , what is the degree measure of the largest angle of the triangle?

Problem 11.3.3

AMC 8

Gilda has a bag of marbles. She gives 20% of them to her friend Pedro. Then Gilda gives 10% of what is left to another friend, Ebony. Finally, Gilda gives 25% of what is now left in the bag to her brother Jimmy. What percentage of her original bag of marbles does Gilda have left for herself?

Problem 11.3.7(additional)

AMC 8

Suppose 15% of $x$ equals 20% of $y$ . What percentage of $x$ is $y$ ?

Problem 11.3.9(additional)

MATHCOUNTS

On Wednesday, $\frac{1}{3}$ of the students in Mr. Short's homeroom had drama practice, $\frac{1}{4}$ of his other homeroom students had band practice. If 6 students had band practice, how many students are in Mr. Short's homeroom?

Contest cornerreal contest problems

Problem 11.3.2

AMC 8

There are 270 students at Colfax Middle School, where the ratio of boys to girls is $5:4$ . There are 180 students at Winthrop Middle School, where the ratio of boys to girls is $4:5$ . The two schools hold a dance and all students from both schools attend. What fraction of the students at the dance are girls?

Problem 11.3.4

AMC 8

All of Marcy's marbles are blue, red, green, or yellow. One third of her marbles are blue, one fourth of them are red, and six of them are green. What is the smallest number of yellow marbles Marcy can have?

Problem 11.3.5

AMC 8

A number of students from Fibonacci Middle School are taking part in a community service project. The ratio of 8th-graders to 6th-graders is $5:3$ , and the the ratio of 8th-graders to 7th-graders is $8:5$ . What is the smallest number of students that could be participating in the project?

Problem 11.3.6

AMC 8

Chloe and Zoe are both students in Ms. Demeanor's math class. Last night they each solved half of the problems in their homework assignment alone and then solved the other half together. Chloe had correct answers to only 80% of the problems she solved alone, but overall 88% of her answers were correct. Zoe had correct answers to 90% of the problems she solved alone. What was Zoe's overall percentage of correct answers?

Problem 11.3.10(additional)

AMC 10

The ratio of $w$ to $x$ is $4:3$ , the ratio of $y$ to $z$ is $3:2$ , and the ratio of $z$ to $x$ is $1:6$ . What is the ratio of $w$ to $y$ ?

(Type the ratio $a:b$ as the fraction $a/b$ .)

Problem 11.3.11(additional)

AMC 8

A store increased the original price of a shirt by a certain percent and then decreased the new price by the same amount. Given that the resulting price was 84% of the original price, by what percent was the price increased and decreased?

Checkpoint for section 11.1

Get 2 of 3 right to master this section. One try each, no hints.

A bag has red and blue marbles in the ratio $3:5$ . There are 40 marbles. How many are blue?

Look at the chain above: $+20\%$ then $-20\%$ does not get back to the start. Now a $50 shirt goes up 10%, then down 10%. What is the new price in dollars?

11.20%

Rate and Work

In this section you will be able to use Work = Rate × Time, and add rates to find how long a team takes.

One pipe fills a tank in 3 hours. A second pipe fills it in 6 hours. Turn both on. Is it 9 hours? 4.5 hours? Something else?

Times don’t add, but rates do. A pipe that fills the tank in 3 hours fills one third of the tank each hour. The other fills one sixth each hour. Together they fill one half each hour, so the tank is full in 2 hours.

Look:to combine workers, change each time into a rate, add the rates, then turn the total back into a time.

Exploreplay first, formulas later

Try this: Turn off Pipe B and press Play. How long does Pipe A take alone? What fraction of the tank does it fill each hour?
Try this: Turn Pipe B back on. Before you press Play, guess: is together faster than 3 hours? Then check where the thick line hits full.
Try this: Turn on Show the formula. Change Pipe B to 3 hours. Two equal pipes: why is the time exactly half?
Try this: Try 4 and 12 hours. Does 4 × 12 ÷ (4 + 12) match the time on the graph?

Name it

Theorem 11.2.1 · Work, Rate and Time

\text{Work} = \text{Rate} \times \text{Time}

Equivalently,

\text{Rate} = \frac{\text{Work}}{\text{Time}} \qquad \text{Time} = \frac{\text{Work}}{\text{Rate}}

Theorem 11.2.2 · Working Together

If one person can do something in $a$ amount of time, and someone else can do it in $b$ amount of time, together they can do it in

\frac{ab}{a+b}

time.

Remark 11.2.3

This not only applies to work. For example, if a problem says 2 faucets take $a$ and $b$ hours to fill a tub, together they can fill a tub in

\frac{ab}{a+b}

hours.

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Example 11.6

Omega Learn

10 workers from a company each working at a constant rate can build 10 houses in 12 years. However, after 6 years, 5 of them retire. The other workers continue working at the same original rate. The project is falling behind, so 2 years later, the company hires 10 more workers who each work twice as fast as the original workers. After how many total years will the 10 houses be complete?

1
After 6 years, what fraction of the 10 houses will be built?

Predict first:
Your prediction

Example 11.7

Omega Learn

Alex, using substitution or elimination, can solve 20 two variable equations in 10 minutes. Bob, who uses the diagonal product method, can solve 20 two variable equations in 2 minutes. How much time will it take both of them working together to solve 60 two variable equations?

1
How many equations does Alex solve per minute?

Predict first:
Your prediction

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Example 11.8

Omega Learn

All experienced workers work at a constant rate and all new workers work at a different constant rate. 12 experienced workers and 6 new workers can build a house in 6 months. 6 experienced workers and 12 new workers can build a house in 9 months. How many months will it take 9 experienced workers and 9 new workers to build a house?

1
Call one experienced worker's rate $e$ and one new worker's rate $n$ (houses per month). What does the first team tell you?

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Example 11.9

Omega Learn

Two rocket scientists Bill Einstein and Albert Nye are building a spaceship, and they will work 6 hours a day for standard pay. It would take Albert a total of 48 hours to build the rocket individually, and it would take Bill a total of 36 hours. The manager can motivate any of them to work extra hours every day by giving them an extra $100 for each hour. The manager only has $1000 to give to Albert and Bill for extra pay. What is the minimum number of whole days that the rocket can be completed?

1
What fraction of the rocket does each scientist build per hour?

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Problem 11.3.13(additional)

AMC 8

When the World Wide Web first became popular in the 1990s, download speeds reached a maximum of about 56 kilobits per second. Approximately how many minutes would the download of a 4.2-megabyte song have taken at that speed? (Note that there are 8000 kilobits in a megabyte.)

Contest cornerreal contest problems

Problem 11.3.12(additional)

MATHCOUNTS

Andre can complete $\frac{5}{6}$ of a job in $\frac{3}{4}$ of the time that it takes Michael to do the whole job. What is the ratio of the rate at which Andre works to the rate at which Michael works? Express your answer as a common fraction.

Checkpoint for section 11.2

Get 2 of 3 right to master this section. One try each, no hints.

11.3

Practice Problems

In this section you will be able to mix ratios, percents and rates on contest problems.

Drying grapes, marbles rolling through pipes, and a final exam to plan for. Each one hides a ratio or a percent. Every problem has three hints if you get stuck.

Want every problem from this chapter on one page? Open the problem set.

Contest cornerreal contest problems

Problem 11.3.8(additional)

AMC 10

Suppose $A > B > 0$ and A is $x\%$ greater than $B$ . What is $x$ ?

(A) $100\left(\frac{A-B}{B}\right)$ (B) $100\left(\frac{A+B}{B}\right)$ (C) $100\left(\frac{A+B}{A}\right)$ (D) $100\left(\frac{A-B}{A}\right)$ (E) $100\left(\frac{A}{B}\right)$

Problem 11.3.14(additional)

MATHCOUNTS

Grapes are 80% water by weight. When a bushel of grapes dries in the sun for two weeks, it loses 50% of its total weight. All of the weight loss is due to the loss of water. After drying for two weeks, what percentage of the grapes is water by weight? Express your answer to the nearest percent.

Problem 11.3.15(additional)

MATHCOUNTS

Ten thousand marbles are released into the top pipe as shown and roll down the pipe system. Anytime the pipe forks, the marbles split in proportion to the cross-sectional areas of the pipes. All pipes have circular cross-sections with diameters as indicated in the figure. How many marbles exit through the bottom, middle pipe?

Problem 11.3.16(additional)

MATHCOUNTS

Brian has earned 65%, 80% and 92% on his three pre-final exams. These exams are not weighed equally: the lowest counts for only 20% of his overall grade, while the other two count for 25% each. If the final exam is the remainder of the overall grade and there are no opportunities for extra credit, what is the highest grade Brian can earn in the class? Express your answer to the nearest whole percent.