Mathcounts: Problem Solving Strategies: for the Beginning Level

Discounts, percentage off questions: 

 Consider the following:

There is a melon on sale for $100 dollars.
The vendor offers to sell it to you at a 20% discount.
How much would you have to pay for the melon? 

In situations such as this, it's better to look backwards at the percentages.
A 20% discount means that there is 100-20, or 80%, of the original price left.
Therefore, you would have to pay $100 x 80% = $100 x 80/100 = 80$ for the melon.

This method can be used for all percentage problems, including more complicated ones, as follows:

The melon vendor has switched to selling strawberries. Each box of strawberries will cost you only $480! After a period of financial devastation, the vendor lowers the price to 40% off. He then offers you an additional 10% off, because you purchased so many melons from him earlier.How much would you have to pay for the box of strawberries?

The 40% off means that only 60% [100% -40%] is left. The additional 10% off means that only 90% [100% - 10%] of the 60% is left. You would have to pay ($480 x 60% x 90%), or $480 x 3/5 x 9/10 = $259.20.

How Consumers Miscaculate Sales Prices : from Consumeraffairs.com

Profits, marked-up, percentage increase questions: 

To make 40% profit, the vendor charges his customer 84 dollars, what is the wholesale price? 

Let the wholesale price be x.
(100% + 40%) * x = 84;     1.4x = 84;      x = 60 dollars

Most students forgot to use 100%, which is equivalent to the original price, on top of 40% and get a real weird answer. Make sure you understand why and how.

This month Mary's monthly income has increased from $2400 to $3000, what is her salary increase?

Two ways to approach this problem:
Solution I: |2400 - 3000| over 2400 = 1/4 = 25% increase
Percentage increase/decrease = |positive difference of the two prices| over the original price

Solution II: (3000/2400) -1 = 25% You find what multiple of the new price versus the original and  
then minus 1 (the original 100%). If it's negative, it's percentage decrease.

More questions to ponder (answers and solutions below):

#1 :You’re walking by a store window and you see a sign that says, “20% off the original price plus an additional 25% off the already reduced sale price.What percentage of the original price do you need to pay? How much is the discount?


# 2: Would you be more likely to buy a product that is 45 % off, or the same product in a store wide 25 % off sale with an additional 25 % off that product?

# 3: A store is offering a promotion; every week, another 50% is taken off current sales prices. Your friend calls you excitedly and says: "Hey, I'll wait for two weeks, since the item will then be free." What's wrong with that statement?


#4: Which is a better deal? 20% off an item that costs 13500 or 10% off and another 10% off of the same item? How much will you save for the cheaper option. 

#5: A computer is on sale for $1260, which is 25% discount off the regular price. What is the regular price? 

#6: A computer's retail price is $1260, which is a 25% markup of the wholesale price. What is the wholesale price?

#7: To make 30% profit, you charge your customer $195, what is the wholesale price?

#8: A store owner offers 20% discounts of a sales item; however, he doesn't want to lose money on the deal. What percentage markup of the wholesale price should he implement so that he's not losing money? 

#9: To make 20% profit, you charge 132 dollars for certain items you sold; however, your friends get 20% discount off the wholesale price. How much do they need to pay? 

#10:  If A is 130% of B and C is 195% of B, what fraction of C is A? 

Answer key and Solutions: 

#1: 0.8 [100% -20%] * 0.75 [100% - 25%] = 0.6 or 60% so you pay 60% of the original price, which is (100% - 60%) or 40% discount. 

#2: After 45% discount, you pay (100% - 45%) or 55% of the regular price. 

However, using the second pricing method you need to pay (100% - 25%) * (100% - 25%) = 0.75 * 0.75 = 0.5625 = 56.25% of the regular price. The first one is cheaper by 1.25%

#3: (100% - 50%) (100% - 50%) = 0.5 times 0.5 = 0.25 = 25% so you still need to pay 25% of the regular price. 
#4: After 20% off, you pay (100% - 20%) = 80% or 0.8 of the regular price. 
After 10% off and another 10% off, you pay (100-10%) * (100-10%) = 0.9 * 0.9 = 0.81 or 81% of the regular price.
Thus, 20% off is a better deal and you save 1% of $13500, which comes up to 135 dollars.
#5: Let the original price be "x" dollars. (100%-25%) * x = 1260;  x = 1260 divided by 0.75 =  
1680 dollars
#6: Let the wholesale price be "x". (100% + 25%) * x = 1260;  x = 1260 divided by 1.25 =                     1008 dollars.
#7: 195 divided by (100% + 30%) = 1.95 divided by 1.3 = 150.
#8: If the store owner offers 20% discount of an item, the customers will pay (100% - 20%) = 80% or 4/5 of the regular price. To not lose any money, let the markup percentage be "x".
4/5 of (100% + x) = 1, x = 25%
#9: 132 divided by 1.2 = 110;    110 * 0.8 = 88 dollars
#10: 130%  over 195% = 130/195 = 2/3

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