We’ve spent a bit of time recently dealing with proportions,particularly proportion word problems. Today, we’re goingto see how the idea of proportions ties into percentages.We’ve spent a bit of time recently dealing with proportions,particularly proportion word problems. Today, we’re goingto see how the idea of proportions ties into percentages.
Better yet, we’re going to learn – relatively quickly – how toeasily handle ANY math problem or real-life situation inwhich percentages play a role.Better yet, we’re going to learn – relatively quickly – how toeasily handle ANY math problem or real-life situation inwhich percentages play a role.
All you have to do today is:
1.Memorize a very simple phrase.1.Memorize a very simple phrase.
2.Learn how to apply it.2.Learn how to apply it.
3.Become really good at re-writing word problems intoshort sentences that simply ask the question you’retrying to find the answer to.3.Become really good at re-writing word problems intoshort sentences that simply ask the question you’retrying to find the answer to.
(Oh, and it’s a really good idea to take some notes today. Seriously.)
You’re not alone if you find the following types of problemsconfusing or difficult:You’re not alone if you find the following types of problemsconfusing or difficult:
You’re not alone if you find the following types of problemsconfusing or difficult:
Despite what you might think, all of the problems shownhere can be solved using the same method. This method isknown by the following phrase:Despite what you might think, all of the problems shownhere can be solved using the same method. This method isknown by the following phrase:
“Is Over Of = Percent Over 100”“Is Over Of = Percent Over 100”
Written in “math,” it looks like this:Written in “math,” it looks like this:
IS = %IS = %
OF 100OF 100
IS = %IS = %
OF 100OF 100
The key to solving percentage problems is identifying theparts within each problem, because ANY problem becomesmuch easier to solve if you can identify the parts.The key to solving percentage problems is identifying theparts within each problem, because ANY problem becomesmuch easier to solve if you can identify the parts.
There are three types of problems you can face:There are three types of problems you can face:
As you read a problem involving percents, recognize thatyou’ll ALWAYS be given two pieces of data; your job is tofind the third. The fourth piece of data – the “100” – isALWAYS present in the equation. In other words, you’renever going to be solving for the value that belongs in the100 spot in the equation, because 100 is always there.As you read a problem involving percents, recognize thatyou’ll ALWAYS be given two pieces of data; your job is tofind the third. The fourth piece of data – the “100” – isALWAYS present in the equation. In other words, you’renever going to be solving for the value that belongs in the100 spot in the equation, because 100 is always there.
(Mr. Laney says it’s probably a really good idea to copy the chart above)
In most situations, the word “what” will show you whichitem in your “is over of” equation is unknown.In most situations, the word “what” will show you whichitem in your “is over of” equation is unknown.
In the first example above, “what percent” indicates that thevariable goes in the “%” position.In the first example above, “what percent” indicates that thevariable goes in the “%” position.
In the second example, “what number is” indicates that thevariable belongs in the “IS” position.In the second example, “what number is” indicates that thevariable belongs in the “IS” position.
In the third example, “of what number” indicates that thevariable belongs in the “OF” position.In the third example, “of what number” indicates that thevariable belongs in the “OF” position.
(Seriously, if you haven’t copied this chart yet, you’re going to struggle)
Let’s look at each of those three problems individually:Let’s look at each of those three problems individually:
Because the phrase “what percent” indicates the variablegoes in the “%” location, all you have to do is decide wherethe 3 and the 4 belong. Read slowly; usually, the location foreach piece of data is given away by the nearest word.Because the phrase “what percent” indicates the variablegoes in the “%” location, all you have to do is decide wherethe 3 and the 4 belong. Read slowly; usually, the location foreach piece of data is given away by the nearest word.
“Three is what percent of 4?”“Three is what percent of 4?”
Once you’ve identified the pieces, place them into yourequation and cross-multiply to solve:Once you’ve identified the pieces, place them into yourequation and cross-multiply to solve:
3 = x3 = x
4 1004x = 300 4 1004x = 300
X = 75; therefore, 3 is 75 percent of 4.X = 75; therefore, 3 is 75 percent of 4.
Let’s look at each of those three problems individually:Let’s look at each of those three problems individually:
Because the phrase “what number is” indicates the variablegoes in the “IS” location, all you have to do is decide wherethe 75% and the 4 belong.Because the phrase “what number is” indicates the variablegoes in the “IS” location, all you have to do is decide wherethe 75% and the 4 belong.
“What number is 75% of 4?”“What number is 75% of 4?”
Once you’ve identified the pieces, place them into yourequation and cross-multiply to solve:Once you’ve identified the pieces, place them into yourequation and cross-multiply to solve:
x = 75x = 75
4 100100x = 3004 100100x = 300
X = 3; therefore, 3 is 75 percent of 4.
Let’s look at each of those three problems individually:Let’s look at each of those three problems individually:
Because the phrase “of what number” indicates the variablegoes in the “OF” location, all you have to do is decide wherethe 3 and the 75% belong.Because the phrase “of what number” indicates the variablegoes in the “OF” location, all you have to do is decide wherethe 3 and the 75% belong.
“Three is 75% of what number?”“Three is 75% of what number?”
Once you’ve identified the pieces, place them into yourequation and cross-multiply to solve:Once you’ve identified the pieces, place them into yourequation and cross-multiply to solve:
3 = 753 = 75
x 10075x = 300 x 10075x = 300
X = 4; therefore, 3 is 75 percent of 4.X = 4; therefore, 3 is 75 percent of 4.
IS = %IS = %
OF 100OF 100
The “Is Over Of” formula works great for word problems.You may have to re-word the problem to help you determinewhich pieces of data belong where, but with a bit of practiceit should become fairly routine:The “Is Over Of” formula works great for word problems.You may have to re-word the problem to help you determinewhich pieces of data belong where, but with a bit of practiceit should become fairly routine:
Carbon makes up 18.5% of the human body by weight. Howmuch carbon is contained in a person weighing 145 pounds?Carbon makes up 18.5% of the human body by weight. Howmuch carbon is contained in a person weighing 145 pounds?
Read slowly; recognize that the % sign is a giveaway that18.5 belongs in the % position. Plus, the 100 is ALWAYSPRESENT. Where does the 145 belong, and where in theequation will you place the variable?Read slowly; recognize that the % sign is a giveaway that18.5 belongs in the % position. Plus, the 100 is ALWAYSPRESENT. Where does the 145 belong, and where in theequation will you place the variable?
IS = %IS = %
OF 100OF 100
Read carefully, and you can place everything properly.Read carefully, and you can place everything properly.
Carbon makes up 18.5% of the human body by weight. How muchcarbon is contained in a person weighing 145 pounds?Carbon makes up 18.5% of the human body by weight. How muchcarbon is contained in a person weighing 145 pounds?
Additionally, you can re-word the word problem into amuch shorter question:Additionally, you can re-word the word problem into amuch shorter question:
What is 18.5% of 145 pounds?What is 18.5% of 145 pounds?
x = 18.5x = 18.5
145 100145 100
100x = 2682.5100x = 2682.5
x = 26.825 poundsx = 26.825 pounds
IS = %IS = %
OF 100OF 100
Let’s try another one:Let’s try another one:
Caitlin made some Valentine’s Day cookies using red and blue M&M’sThe recipe called for 40% of the M&M’s to be blue. If she uses a total of80 M&M’s, how many RED ones does she need?Caitlin made some Valentine’s Day cookies using red and blue M&M’sThe recipe called for 40% of the M&M’s to be blue. If she uses a total of80 M&M’s, how many RED ones does she need?
We know that 40% of the M&M’s are blue; that means the other60% are red, and the problem is asking us to determine howmany red M&M’s we need. After reading the word problem, wecan re-word it into a short, essential question:We know that 40% of the M&M’s are blue; that means the other60% are red, and the problem is asking us to determine howmany red M&M’s we need. After reading the word problem, wecan re-word it into a short, essential question:
What is 60% of 80 total M&M’s?What is 60% of 80 total M&M’s?
IS = %IS = %
OF 100OF 100
Caitlin made some Valentine’s Day cookies using red and blue M&M’sThe recipe called for 40% of the M&M’s to be blue. If she uses a total of80 M&M’s, how many RED ones does she need?Caitlin made some Valentine’s Day cookies using red and blue M&M’sThe recipe called for 40% of the M&M’s to be blue. If she uses a total of80 M&M’s, how many RED ones does she need?
What is… 60%... of 80 total M&M’s?What is… 60%... of 80 total M&M’s?
x = 60x = 60
80 10080 100
100x = 4800100x = 4800
x = 48x = 48
48 of Caitlin’s M&M’s need to be red.48 of Caitlin’s M&M’s need to be red.
IS = %IS = %
OF 100OF 100
Try your hand at this one:Try your hand at this one:
In a typical year, your parents spend $4,800 on food. If you eat $1,500 ofthis total, what percentage of your family’s food are you eating?In a typical year, your parents spend $4,800 on food. If you eat $1,500 ofthis total, what percentage of your family’s food are you eating?
Take a look at the question. You are being asked to find thepercentage, so clearly the variable will go in the “%” position.Now… where does the $1,500 and $4,800 go? Re-word the probleminto a short, essential question:Take a look at the question. You are being asked to find thepercentage, so clearly the variable will go in the “%” position.Now… where does the $1,500 and $4,800 go? Re-word the probleminto a short, essential question:
$1,500 is what percent of $4,800?$1,500 is what percent of $4,800?
IS = %IS = %
OF 100OF 100
In a typical year, your parents spend $4,800 on food. If you eat $1,500 ofthis total, what percentage of your family’s food are you eating?In a typical year, your parents spend $4,800 on food. If you eat $1,500 ofthis total, what percentage of your family’s food are you eating?
$1,500 is…. what percent…. of $4,800?$1,500 is…. what percent…. of $4,800?
1500 = x1500 = x
4800 100 4800 100
4800x = 1500004800x = 150000
x = 31.25x = 31.25
You are eating 31.25% of your family’s food. You pig.You are eating 31.25% of your family’s food. You pig.
It takes a bit of practice, but once you get the hang of it youshould be able to handle any percentage problem thrown atyou. Remember the three steps:It takes a bit of practice, but once you get the hang of it youshould be able to handle any percentage problem thrown atyou. Remember the three steps:
1.Memorize a simple phrase (“is over of = % over 100”).1.Memorize a simple phrase (“is over of = % over 100”).
2.Learn how to identify the pieces of data in the problem.2.Learn how to identify the pieces of data in the problem.
3.Re-write word problems into short, essential questions.3.Re-write word problems into short, essential questions.
One final thought:One final thought:
CHECK YOUR ANSWER FOR REASONABLENESS!CHECK YOUR ANSWER FOR REASONABLENESS!
If it isn’t reasonable, you’ve got your data in improper places. Trymoving some things around in your equation in order to come up with amore logical answer. A five-second check to ask yourself “does this makesense” will usually eliminate ALL errors. If it isn’t reasonable, you’ve got your data in improper places. Trymoving some things around in your equation in order to come up with amore logical answer. A five-second check to ask yourself “does this makesense” will usually eliminate ALL errors.