A state department of education, as a question on the public-release form of a statewide standardized test, released a question to the public in which eighth-grade students have to determine how many students are now in a school that had 80 students last year if the school has 10 percent more students this year than last year.
First things first. What is 10% of 80? Well, we could do it the long way and multiply 80 × 10%, which means 80 × 0.10, since “percent” means “out of 100.” That is why we should write 10% as 10 hundredths, or 0.10.
That comes to 8, but we could have saved ourselves a step with this problem. We know that 10 percent of any number is just the number that you get when you move the decimal point one place to the left:
And if 10 percent of 80 is 8 and the problem says there are 10% more students this year than last, that means there are 8 more students this year than there were last year. Last year, the school had 80 students, and this year we have 8 more students.
Therefore there are 88 students at the school this year, which is 10 percent more than there were last year.
This problem is similar to finding the tax on a sale. For instance, let’s say your double cheeseburger costs $1 (that is the price for a double cheeseburger at McDonald’s and at similar fast-food restaurants) and you buy two of them along with a Diet Coke that costs $1.50, bringing your total to $1 + $1 + $1.50 = $3.50.
If the tax rate where you live is 8 percent, how much will you pay?
As above, we put first things first: What is 8% of $3.50? Since 8 percent means “8 hundredths,” we multiply the total cost of the food (3.50) by 0.08. And once we know how much the 8-percent tax will be, we add it to the 3.50, shown below:
That is to say, .28 is 8 percent of 3.50. Tax always adds to the actual purchase price. In other words, first we multiply to find the amount of just the tax, and then we add that tax amount to the purchase price for our cheeseburgers and Coke.
If you would like additional problems to help you master this skill, go to our online card catalog at VoxLearn.org, select one of the math collections, and enter the search terms “tax tips commissions”.
Try a few on your own
Another way to ask this type of question is to determine how much of an increase occurred from one year to the next.
1. The results for the National Assessment of Educational Progress (NAEP) show that 36 percent of the eighth graders in Maryland achieved a score of proficient or higher on the mathematics portion of the test in 2007, while only 30 percent of the state’s eighth graders achieved this mark on the 2005 test. What was the percent increase in the proportion of students who achieved proficient or higher scores in 2007 compared to 2005?
2. If your family’s dinner bill at a fancy restaurant comes to $90 for just the price of the food and your tax rate is 8 percent, how much of a tip should you leave and what will be your total expenses for the visit to the restaurant?
3. You open a savings account at a bank that pays 2.5% interest annually on whatever money is in the account. If you deposit $450 in the account when you open it, how much money will you have after two years?
The Answers
(1) NAEP increases for Maryland eighth graders
First, the actual increase from 2005 to 2007 went from 30 percent of the students who took the test to 36 percent. That increase of 6 percentage points in the proportion, though, represents a 20 percent increase in the proportion from 2005 to 2007.
You are given the following information:
Now, if we were looking at an increase from 100 to 106, that would be a simple 6-percent increase, because we started at 100. But here we’re starting at 30.
Think of it like this: a number that increases from 2 to 4 doubled. If a value doubles, it represents a 100-percent increase. If a number triples, it represents a 200-percent increase. And so on.
What we have here is an increase of 6, starting at 30. As you see in the 100- and 200-percent increase examples, the actual percent of increase will be higher than the increase in the numbers themselves, provided the number you’re starting with is lower than 100.
This increase of 6 percentage points in the proportion of students who scored proficient or above is one-fifth of 30, and since we started at 30 in 2005, we added one-fifth of that 30 (which is 6) to the 30, making 36, the value for 2007.
One-fifth of something is equivalent to 20 percent of that something. Therefore, the proportion of eighth graders in Maryland scoring proficient or higher on the mathematics portion of the NAEP increased by 20 percent in 2007, compared to its 2005 value.
(2) Tax and Tip on an expensive dinner
The food cost was $90, and with an 8-percent tax rate, we will pay $90.00 × 0.08 = $7.20 in taxes.
For the tip, a typical amount is 15 percent. Let’s use that, assuming your waiter or waitress was adequate. For now, 15 percent of $90 = $90.00 × 0.15 = $13.50. Not too shabby.
Adding it all up to find our total cost for the restaurant, we get a total of $90.00 + $7.20 + $13.50 = $110.70. Wow.
(3) Interest on a savings account
There is actually a formula for figuring this out, but we’re going to deal with the problem as it applies to the eighth-grade math in the Illinois curriculum.
Working out the math, we see that after one year, we will earn 2.5% on a balance of 450.00. What is 2.5 of 450.00? First, we multiply our principal amount ($450) by the interest rate (2.5%). That gives us $11.25, which will be the interest paid after the first year. We add that interest to the principal to give us a principal plus interest of $461.25 after the first year:
In the table, then, our New Balance after the first year is $461.25:
Repeating exactly the same procedure for the second year (except that our interest is now computed on a balance of $461.25 instead of $450), we find that the amount of interest paid in the second year will be 461.25 × 0.025 = 11.53125. Of course, your bank will round to the nearest penny, making the amount they actually pay you 11.53 (the extra 1/8 of a penny just kind of disappears on the bank’s books, I think).
Adding the second year’s interest (11.53) to the balance when the year began (461.25), we get a new balance of $472.78. You’re rich.
Illinois Alignment
Grade Level 8
Illinois Learning Standard 6.C.3a (middle school) Select computational procedures and solve problems with whole numbers, fractions, decimals, percents and proportions.
Illinois Assessment Framework 6.8.18 (8th grade) Solve number sentences and problems involving fractions, decimals, and percents (e.g., percent increase and decrease, interest rates, tax, discounts, tips).