In Chapter 5, you learned how to find the percent of a number by making a diagram to
relate the part to the whole and find the desired portion. This calculation is fairly
straightforward if the percent is a multiple of 10, like 40%, or can be thought of as a
fraction, like
= 25%. However, it can be more challenging if the percent is something
like 6.3% or 84.5%.
Today you will connect what you have learned previously about the relationship between
distance, rate, and time to the idea of scale factors. You will learn how to use a scale
factor to find the corresponding lengths of similar figures. This idea will add a powerful
new tool to your collection of problem-solving strategies that will help you to calculate
percents.
7-16. Dana is training for a bicycle race. He can ride his bike 25 miles per hour.
One day, when he had been riding for
of an hour, he had to stop and fix a flat
tire. How many miles had he ridden when he stopped? The diagram below may be
useful.
7-17. Matt thought about problem 7-16 and drew the diagram at right. Look at Matt’s
drawing and decide how he is thinking about this problem.
A. Write an equation that uses the scale factor to find x.
B. What connection is Matt making between finding a distance using the rate and
time (as you did in problem 7-16) and using a scale factor with similar
figures? How are the situations alike and how are they different?
7-18. In the two previous problems,
is used in two ways: first, as time in the rate
problem
, and second, as the scale factor in the similar triangle problem
used to find three-fifths of 25 miles. Both of these situations resulted in an equivalent
calculation:
. How else could this be written?
a. Using the portions web shown at right, work with your team to find two other
ways to write the equation
be
b.
. For example, one way might
.
If you did not already find it, what percent would be equivalent to ? Use this
percent to write a statement in words and symbols that is equivalent
to
.
c. Use the idea of scaling to find the following values. Write an expression using
either a fraction or a percent, and then find the result.
i. 90% of 25 miles
ii. 8% of $75
iii. 25% of 144
7-19. Josea went out to dinner at an Indian restaurant. The total bill was $38. She
wanted to leave a 15% tip.
a. If you use the idea of scaling to find the tip amount, what would she need to
multiply by? As you talk about this with your team, consider:
i. How could you represent this multiplier as a fraction?
ii. How could you represent it as a decimal?
iii. Does it make a difference which representation, fraction or decimal, you
use to solve this problem?
iv. Which do you think will be easier?
b. How much should Josea leave for the tip? Show your calculations.
c. If Josea changes her mind and wants to leave a 20% tip instead, how much will
this be?
7-20. While shopping for a computer game, Isaiah found one that was on sale for 35%
off. He was wondering if he could use
as a multiplier to scale down the price to
find out how much he would have to pay for the game.
A. If Isaiah uses
as a scale factor (multiplier), will he find the price that he will
pay for the game? Why or why not?
B. There is scale factor (multiplier) other than 35% that can be used to find the sale
price. What is it? Draw a diagram to show how this scale factor is related to
35%. Label the parts of your diagram “discount” and “sale price” along with the
relevant percents.
C. How much will Isaiah have to pay for the game if the original price is $40? Show
your strategy.
Quartiles and Interquartile Range (IQR)
Quartiles are points that divide a data set into four equal parts (and thus, the use of the
prefix “quar” as in “quarter”). One of these points is the median, which you learned
about in Chapter 1, since it marks the middle of the data set. In addition, there are two
other quartiles in the middle of the lower and upper halves: the first quartile and the
third quartile.
Suppose you have this data set: 22, 43, 14, 7, 2, 32, 9, 36, and 12.
To find quartiles, the data set must be placed in order from smallest to largest. Then
divide the data set into two halves by finding the median of the entire data set. Next find
the median of the lower and upper halves of the data set. (Note that if there are an odd
number of data values, the median is not included in either half of the data set.) See the
example below.
The interquartile range (IQR) is one way (along with range) to measure the spread of
the data. Statisticians often prefer using the IQR to measure spread, because it is not
affected much by outliers or non-symmetrical distributions. The IQR is the range of the
middle 50% of the data. It is calculated by subtracting the first quartile from the third
quartile. In the above example, the IQR is 26 (34− 8− 26.)