Line graphs are generally used to show the change in a quantity over time. A runner might record her times at each daily practice and create a line graph to help her see the pattern of improvement, or to see if there was a particular time when her times were slower or faster than usual.

To create a line graph, you need data that has been recorded. Each entry should include a date or time and a value. Let’s suppose that runner recorded her times in seconds for the 200 meters every day in April. Here is her record for the first 10 days.

Day 1 2 3 4 5 6 7 8 9 10

Time 22.71 22.68 22.50 22.43 22.37 22.19 22.04 21.97 21.91 21.89

The horizontal scale should show the day numbers, in this case up to 10, equally spaced along the axis. The vertical scale must go high enough to record all the times, so up to about 23 seconds.

Then place a dot in line with each day’s number at a height that corresponds to that day’s time in seconds on the vertical axis. Finally, connect the dots, day 1 to day 2 to day 3 and so on, using straight line segments.

You may find that your first attempt is not satisfactory. If you labeled your vertical scale from 0 to 23 seconds, the graph may look very flat. The runner’s times are only changing by a fraction of a second at a time.

A broken scale is a good idea here. Here’s what the same graph looks like if you use a broken scale, showing only 21.4 to 22.8 instead of 0 to 23. It’s much easier to see the changes in the runner’s times.

The line graph shows the monthly net profit statement for Amy’s

Antiques from January to December.

In which month was the net profit the same as the net profit in February?

What is the total net profit for the first quarter of the year (that is, January, February, and March combined)?

Which month shows the same increase in net profit, when compared with the previous month, as was shown in April when compared with March?

Which pair of consecutive months shows a combined net profit of exactly $8,800?

Which month accounts for approximately 5% of the total yearly net profit?

game. Notice that 24 out of the 108

cards are yellow.

Heart | 24 |

Yellow | 24 |

Purple | 24 |

Green | 24 |

Wild Draw | 10 |

Extra Cards | 2 |

You can compare these two numbers

by using a ratio. Recall that the ratio

that compares 24 to 108 can be

written in several ways.

24 to 108

24:108

24 out of 108

\(\frac{24}{108} \)

A common way to express a ratio

is as a fraction in simplest form.

\(\frac{24}{108}=\frac{2}{9}\)

So, \(\frac{2}{9}\) of the cards are yellow.

The ratio can also be expressed as 2 to 9, 2:9, or 2 out of 9.

If the two quantities that you are comparing have different units of

measure, the ratio is called a rate. For example \(\frac{180 miles}{3 hours}\) compares the

number of miles traveled to the number of hours the trip took.

The word about tells you that an exact answer is not needed.

So, you can estimate the answer. When you estimate with percents, you should round the percent to a fraction that is easy to multiply. The chart shows some commonly-used percents and their fraction equivalents.

hundredths, the probability can also be expressed as the ratio

.

This means that the player makes a free throw 75 out of 100 times. In

this lab, you will explore ratios and probability.

A player makes a basket every 50 out of 80 times. What is the probability that he will make a basket on his next attempt? Express the probability as a decimal.

One way to determine whether two ratios form a proportion is to find their If the cross products of two ratios are equal, then the ratios form a proportion.

Advertising According to an advertisement for Brand X toothpaste, 8 out of 10 dentists prefer Brand X. There are 150 dentists in a certain city.

Predict how many of them prefer Brand X.

Number Sense A number multiplied by itself is 676. What is the number?

School At the Science Festival’s bridge-building competition, Juliet came in second, and Daniel finished behind Pedro. Keela finished ahead of Pedro, and Reynelda won first place. In what order did they finish?

Recreation Robin purchased a tent for camping. Each side of the four sides of the tent needs 3 stakes to secure properly to the ground. How many stakes should there be in the box?

]]>as fractions and to solve proportions by using cross

products. Then, using their knowledge of ratios, students

find actual lengths from scale drawings. Students explore

the relationship among fractions, decimals, and percents

as well as find the percent of a number. Students also

learn to solve problems by drawing a diagram.

You’ll learn

to express ratios and rates as fractions,

to solve proportions by using cross products,

to solve problems by drawing a diagram,

to find actual length from a scale drawing,

to express percents as fractions and as decimals, and

to find the percent of a number.

Knowing how to express ratios as fractions can help you determine better buys in a grocery store.

If the two quantities that you are comparing have different units of measure, the ratio is called a rate.

Write each ratio in three different ways.

1. 9 pairs of boots out of 16 are yellow.

2. 13 bikes out of 16 bikes have combination locks.

3. 18 out of 29 centimeter cubes are pink.

4. 11 brownies out of 15 brownies contain walnuts.

5. 23 out of 25 ants are red.

6. 7 out of 19 days were cloudy.

Geography Refer to the map.

a. Write the ratio that compares the number of states in the Southwestern States region to the total number of states.

b. Write the ratio that compares the number of states in the Midwestern States to the total number of states.

a number greater than one. The base of an exponential decay function is a number between 0 and 1.

Exponential functions are frequently used to model the growth or decay of a

population. You can use the y-intercept and one other point on the graph to write the equation of an exponential function.

POPULATION

Every ten years, the Bureau of the Census counts the number of people living

in the United States. In 1790, the population of the U.S. was 3.93 million. By 1800, this number had grown to 5.31 million.

Write an exponential function that could be used to model the U.S. population y in millions for 1790 to 1800. Write the equation in terms of x, the number of

decades x since 1790.

Assume that the U.S. population continued to grow at that rate. Estimate the

population for the years 1820, 1840, and 1860. Then compare your estimates

with the actual population for those years, which were 9.64, 17.06, and

31.44 million, respectively.

RESEARCH Estimate the population of the U.S. in 2000. Then use the Internet

or other reference to find the actual population of the U.S. in 2000. Has the

population of the U.S. continued to grow at the same rate at which it was

growing in the early 1800s? Explain.

COMPUTERS

If a typical computer operates with a computational speed s today, write an

expression for the speed at which you can expect an equivalent computer to

operate after x three-year periods.

Suppose your computer operates with a processor speed of 600 megahertz and you want a computer that can operate at 4800 megahertz. If a computer with that speed is currently unavailable for home use, how long can you expect to wait until you can buy such a computer?