Task
Katie and Margarita have \$20.00 each to spend at Students' Choice book store, where all students receive a 20% discount. They both want to purchase a copy of the same book which normally sells for \$22.50 plus 10% sales tax.
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To check if she has enough to purchase the book, Katie takes 20% of \$22.50 and subtracts that amount from the normal price. She takes 10% of the discounted selling price and adds it back to find the purchase amount.
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Margarita takes 80% of the normal purchase price and then computes 110% of the reduced price.
Is Katie correct? Is Margarita correct? Do they have enough money to purchase the book?
IM Commentary
This purpose of this task is to help students see two different ways to look at percentages both as a decrease and an increase of an original amount. In addition, students have to turn a verbal description of several operations into mathematical symbols. This requires converting simple percentages to decimals as well as identifying equivalent expressions without variables. In particular the student sees that 110% of an amount is the same as adding 10% to the original amount. (This is an extension of 6EE.4.)
Solution
Katie’s method: Katie first subtracts 20% of the original price from the original price:
$$22.50 − (0.20(22.50)) = 22.50 − 4.50 = 18.00.$$
Next she takes 10% of this new amount and adds it back, so
$$18.00 + (0.10(18.00)) = 18.00 + 1.80 = 19.80.$$
Margarita’s method: Margarita first computes 80% of the original price:
$$(0.80)22.50 = 18.00.$$
Next, she computes 110% of the new amount:
$$(1.10)18.00 = 19.80.$$
The two methods are both correct and the students both have enough money to purchase the book. If we look more carefully, we can see why.
First, consider Katie's method again:
Using the distributive property, we see that subtracting 20% is the same as multiplying by $(1 - 0.20)$:
$$22.50 − (0.20(22.50)) = (1 - 0.20)(22.50) $$
Multiplying by $1-0.20 = 0.80$ is the same thing as finding 80 percent.
Also, adding 10% is the same as multiplying by $(1 + 0.10)$:
$$18.00 + (0.10(18.00)) =(1+0.10)(18.00).$$
Multiplying by $1+0.10 = 1.10$ is the same thing as finding 110 percent.
Since these are accomplished by multiplying one number after the other, we can combine everything together:
$$(1 + 0.10)(1 - 0.20)(22.50) = (1.10)(0.80)(22.50)$$
Katie’s method is what most students use when first learning to think about such problems, building up the answer bit-by-bit. Margarita’s method illustrates that reducing a number by a certain percent is equivalent to multiplying by a decimal between 0 and 1 and increasing a number by a certain percent is equivalent to multiplying by a decimal greater than 1.
