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Math Lecture Notes
Numbers on the Page

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Some math problems throw lots of numbers at you. Careful organization is needed to deal with a bunch of information before any formulas are used.

three carpet cleaning options

1. All the rooms in the floor plan below have ceilings 7' 6" high. The bedrooms and living room are carpeted. The bathroom, kitchen, and dining room have vinyl floors. (a) At $3 per square foot, what is the cost of replacing all the vinyl flooring? (b) Which of the three carpet cleaning services is the best buy, and how much will it cost? (c) If you need to paint the three non-closet walls of the first bedroom, what is the area that you will paint? (The balcony has a plastic "fake grass" surface. The closets are carpeted, but do not need cleaning. The T-shaped hallways has tile. Those areas are not part of our problem.)

two bedroom apartment floorplan

Some graphs are strange because a single item appears multiple times, hidden in different places.

2. Gina's mother keeps pestering her to "shop for a husband while you are young and pretty." Gina's friends assure her that this advice is outdated, and Gina can focus on her career at least until the age of 30 if not 35. Use the graph of census data to decide who is correct. (Hint: a woman who is 35 years old in 2014 appears in four different places on this graph!)

2014 Never Married Census Data graphed by Dalrock

Some problems seems simple, but do not fit our natural intuition. The numbers rush into the wrong places in our heads!

gift certificate

3. As a birthday present, Janice receives a $50 gift certificate to her favorite restaurant. She invites two friends to join her for dinner there. She expects great service and plans to pay a 15% tip. Restaurant sales tax where she lives is 10%. What is the most she can pay for the food if she wants the gift certificate to also cover everyone's tip and tax?

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Subtract or Divide

Absolute and Relative Change

Absolute Change (Subtract)

Imagine a medicine that can reduce one of your family member's cancer risk from 44 cases among 10,000 people down to 11 cancer cases among 10,000 people. The medicine has some bad side effects. Is the reduction in cancer risk worth suffering these side effects?

We could use subtraction to find the absolute change in risk.

Example 8

The old risk is 0.44%. The new risk is 0.11%.

0.44% − 0.11% = 0.33%

We could say that with the medication the risk is reduced by 0.33%. (Only a third of one percent? That does not sound like much.)

Relative Change (Divide)

We could also use a percent change to measure how much less likely is the new risk. Like all percent changes, this is a ratio comparing change to original. (The funky part of this problem is how the change and original amount are both percentages.)

Using a percent change to measure "less likely" is a relative change.

Example 9

The old risk is 0.44%. The new risk is 0.11%. Subtracting tells us the change is 0.33%. Then we do change ÷ original.

0.33% ÷ 0.44% = 0.75 = 75%

We could say that with the medication the occurrence of cancer is 75% less likely than before. (That sounds impressive!)

We could use a ratio to talk about whether the new risk is as likely as the old risk. Since this is another ratio, it is also called a relative change.

Example 10

The new risk is 0.11%. The old risk is 0.44%.

0.11% ÷ 0.44% = 0.25 = 25%

We could say that with the medication the occurrence of cancer is only 25% as likely than before. (That still sounds impressive!)

The moral of the story is to pay attention (especially when dealing with small numbers) to whether a speaker is using an absolute change or a relative change. The former made the medicine sound like it is probably not worth the risk of its side effects. The latter made the medicine sound amazing.

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Missing Link in a Chain Discount

Absolute and Relative Change

What about if we know the single equivalent discount rate, but are missing one of the links?

Example 6 (Starting with the Single Equivalent Discount Rate)

A toy originally selling for $100 is put on sale at 15% off. A store-wide seasonal sale cuts the price another 10%. The store expects the toy will only sell if the price is lowered to $60. What additional discount should be applied with a special coupon?

$100 × 0.85 × 0.9 × r = $60

           $76.5 × r = $60

                  r ≈ 0.78 = 78%

So the missing link has 78% remaining.

The store should apply 100% − 78% = 22% more discount with the coupon.

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More Practice

Flu Virus

51. The medicine trastuzumab, which fights breast cancer in women who already have breast cancer, was popularized because of a certain study. In the control group of 1,700 women, 34 died. In the group treated with trastuzumab, 23 of 1,643 women died. What percentage of the women in the control group died? What percentage of the women in the treated group died?

52. Continuing the previous problem, what was the absolute change (subtraction) in risk? What was the relative change (percent change) in risk?

53. If you were in charge of publicity for this drug, what type of claim could you truthfully make about the medicine?

54. Trastuzumab also has some dangerous side effects. Most notably, 40% of the women who take it develop flu-like symptoms, 7% develop mild heart problems, and 5% suffer a stroke or severe heart failure. If you were trying to discredit trastuzumab (perhaps concerned about the side effects and trying to convince a family member with breast cancer not to take the medicine) what type of claim could you truthfully make about the medicine?

51. For the control, group, 34 ÷ 1,700 = 0.02 = 2%. For the treated group, 23 ÷ 1,643 = 0.0139 ≈ 1.4%.

52. The absolute percent change is 2% − 1.4% = 0.6%. The relative percent change = change ÷ original = 0.6% ÷ 2% = 0.3 = 30%.

53. You could say, "People that take this drug are 30% less likely to die from breast cancer."

54. You could say, "People that take this drug only have their cancer risk reduced by less than 1%, but have a 5% chance to suffer side effects of stroke or severe heart failure."

19. Geoffrey works at a candy store. He knows from past years' experience that after Valentine's Day he needs to reduce the prices of the special $30 chocolate boxes down to $18 to clear out that inventory. He uses a store-wide sale of 10%, hoping that will attract customers. He also distributes a coupon that further discounts the sale price of those expensive chocolate boxes. What percent discount is needed on the coupon?

19. The discount chain equation is:
$30 × (1 − 0.1) × r = $18.

Combine the first two numbers to get $27 × r = $18.

Divide both sides by 27 to get r = 67%.

Since that is what should remains after the coupon decreases the price, the discount should be 100% − 67% = 33%.

Random Homework Problems


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