Percents: Key Concepts and Connections

[Rescued from my old blog.]

Paraphrased from a homeschool math discussion forum:

“I am really struggling with percents right now, and feel I am in way over my head!”

Percents are one of the math monsters, the toughest topics of elementary and junior high school arithmetic. Here are a few tips to help you understand and teach percents.

Basic Concept: Percent Means Hundredths

Related topics: Fractions, decimals, converting between them.

Percents are fractions in disguise. The denominator of a percent fraction is always 100, and we use the symbol as a short-cut rather than writing the denominator each time. I usually point out, as a memory aide, that the symbol % is made of the fraction bar and the two zeros in 100. If students can understand that percents are fractions in disguise, they will be able to build on their knowledge of fractions in dealing with percents. They aren’t learning something entirely new; it is just a new way of writing something they have already learned.

Many times, we find that percent problems are easier to do if the percent is changed to a fraction or decimal. That moves the problem out of the less familiar topic onto more comfortable ground. When you see the textbook multiply by 100 or divide by 100, and you aren’t sure where the 100 came from, they are probably converting from a fraction to a percent or from a percent to a decimal. Watch for this, and help your student learn how to do it. Students need to be able to move easily back and forth between all three ways to represent the number.

Basic Concept: A Percent Is a Fraction that Compares a Part to a Whole

Related topics: Ratio, proportion, defining a unit.

Percent fraction = part / whole.

The part is the number that is being compared to another number. The whole is the number we are comparing to, the number that represents 100%. Because in some cases, the part can be larger than the whole, many textbooks prefer the term base for the number that represents 100%. I don’t like to introduce totally new words when I am describing a basic concept, however, so I stick with the part/whole phraseology, while gradually weaving the term base into my discussion. Even in high school, I will still be reminding my students that the base is the number being treated as the whole in this comparison. (Just like I’m still reminding them that the denominator is the bottom number of a fraction and the perimeter is the distance around the rim of a shape.)

We need to be careful in determining which is the part, the number being compared, and which is the whole or base, the number to which we are comparing. Especially when the problem involves a percent increase or decrease, or profit or loss, or a part of a portion of the original amount, this can get very confusing. In complex story problems (and percent problems in Singapore math do get complex), the base may change in the course of the problem. The base for one percent may not be the same as the base for another percent — and then it is like dealing with fractions that have different denominators. You cannot just add up the percents and think that’s your answer, if the bases are different. But if the student can identify the part and the base, he has done the most challenging part of the problem. The rest is merely number-crunching.

I like to use the percent proportion to solve problems:

part / whole = percent / 100

The part is related to the whole base in the same way that the percent number is related to 100. Unfortunately, students often get confused when cross-multiplying, so many textbooks avoid teaching the proportion. Singapore Primary Math 6 uses the proportion on page 47, but it also used the equation:

(part / whole) x 100 = percent

Then the book continues to switch back and forth between these forms with little explanation of how they are connected. If you look closely, you will see that this equation is the same as the percent proportion, but with half of the cross-multiplying done already. Sometimes Singapore math can be nearly self-teaching, but this is a case where your student may need extra help to recognize that these two equations are the same.

Basic Concept: “Of” Means Multiply

Related topics: Multiplication of fractions, story problems, translating from English to math symbols and vice versa, inverse operations.

I will have taught this translation back when I was teaching fractions, if not before, so this is a chance to review and reinforce the concept. If the student knows that the word of means to multiply, he will be able to translate a percent problem into an equation. First, he must restate the problem as a simple question like:

What Is 75% of 160?

The word what is the unknown value, is means the equal sign, percent means hundredths, and of means multiply. The equation becomes:

something = (75 / 100) x 160

It is a little more difficult when the question has the unknown in the middle, such as:

24 is what percent of 96?

Translating into symbols would give:

24 = (something / 100) x 96

Then the student has to know how to use inverse operations to solve the equation, dividing by 96 and multiplying by 100. In cases like this, the percent proportion is easier to work with than the equation.

Inverse operations is a very basic concept which connects to almost everything in mathematics. Students usually don’t think much about inverse operations until they get to algebra. Primary Math 6 uses bar diagrams to help students work through story problems, in an attempt to make the inverse relationships easier to see. I am not sure the text is successful in this, which may be why so many people struggle with this section on percents.

I think I would rather teach my students to manipulate the basic percent proportion. Yes, it is purely abstract, but students at this stage should have a solid foundation in fractions and be ready to handle abstract concepts like this. If the student can understand how to use inverse operations, then that one simple formula will solve almost every percent problem he meets. If not, then he is stuck with memorizing three different equations for the three different situations — unknown part, unknown base, or unknown percentage. I have seen math textbooks which taught percents that way. What a useless hassle!

Basic Concept: How to Find Percent Values Mentally

Related topics: Money, shopping.

This may be less important than the other basic concepts I’ve mentioned, at least for future work in math class, but it is a useful real-life skill. As I see it, the key percent values for mental calculation are:

50% = 1/2 of the base = base ÷ 2

25% = base ÷ 4

10% = base ÷ 10

1% = base ÷ 100

These are usually the easiest percent values to find. Another helpful approximation is:

33% ~ base ÷ 3

When Singapore math talks about solving problems by the unitary method, it just means to find out what 1% of your base is, and then use that to figure out whatever else you need to know. I prefer to call this the Think “One” Method (hat tip to Ed Zaccaro in his book, Becoming a Problem Solving Genius). Unit means one, so the unitary method focuses on using the easy-to-work-with number one. Once you know what 1% is, you can find any other percent just by multiplying.

1% = base ÷ 100

So to find, say, 85%:

85% = 85 x 1% = 85 x (base ÷ 100)

A student who is good at mental math may also be able to calculate:

5% = half of 10%

35% = 1/4 + 1/10

4% = 1% x 4

or tougher percents like: 96% = 100% – 4%

Percents are indeed a mathematical Jabberwock, with jaws that bite and claws that catch many a careless student. But even a monster like percents will fall to a well-swung vorpal sword. With a good understanding of fractions and plenty of practice, your student will be able to forge his way fearlessly through the tulgey wood of arithmetic.

20 thoughts on “Percents: Key Concepts and Connections

  1. Hi, Jordan!

    I don’t understand your question. Do you mean you have a number, and you want to know what percentage it is of 75? Or do you mean that you want to find out what a certain percent of 75 would be? And is there a specific reason why you don’t want to use division?

    Most percent calculations require division, because percents are fractions. And all fractions are division problems. The line in a fraction means “divided by”, and every fraction is equal to its top number divided by its bottom number.

  2. hi

    i have a final for school coming up and it is worth 20% of my grade.
    i want to know, if i have a, lets say 90% in the class, what will be my grade after the final depending on what i get on the final.


  3. There are a couple of ways that you can look at this. First, if the final is 20% of your grade, that means your other points make up 80% of your grade. That means if you add 20% of the final to 80% of the other points, you would get 100% of the score for that class:

    20 \% \times \left(Score \: on \: final \right) + 80 \% \times \left(Current \: average \right) = Grade

    Another way to look at it is that 20% is 1/5 of the class grade. It is as if you have 5 tests total, and on each of the other 4 tests you got ________ (fill in the blank with your current average, which you said is 90%). The fifth test is your final, and you need to average them all together:

    \frac{4 \times \left(Current \: average \right) + \left(Score \: on \: final \right)}{5}   = Grade

  4. hi i have a question on inverse operations how do you solve a promblem with a fraction and a decimal or a fraction with a fractions?

  5. Ritchie — you need to figure out what you can multiply or divide everything in the equation by to get from whatever-amount-of-x you have, to 1x.

    So if you have 0.5x + 1/3 = 1 (it has to equal something!) what can you do to 0.5x to change it into 1x –> multiply by 2. So you multiply everything by 2 and get x + 2/3 = 2.

    For 1/5, what can you multiply that by to get 1?

  6. Hi, Richie and Dennis.

    “Inverse operations” aren’t really something that you solve. They are more like “opposites” or “partners” to each other. Math operations come in pairs, where each is the inverse of the other. Some pairs are:
    * addition and subtraction
    * multiplication and division
    * powers and roots

    So to do an inverse operation with a fraction, you just do it however you would normally work with a fraction. You add the fraction, or subtract it, or multiply it, or whatever you need to do, according to the rules you have learned for adding or subtracting or multiplying fractions.

  7. how can we change 182.5 into percentage and how can we tell this to in grade 8`s students.actually perecnt means per hundredth it exceed more than hundrer.

  8. A percent number is meaningless by itself. It has to be a percent of something. If you want to change 182.5 into a percentage, then you have to first fill in the blank: “182.5 is what percent of _____?”

    The standard rule for changing a number to a percent is to convert it into hundredths: “Move the decimal point to the right 2 spaces.” That rule will tell you what percent that particular number is of the number one. So the number 182.5 is 18,250% of the number one.

  9. hi
    thats right thatwhen one is going to change any number into percentage just like we have to change 182.5 into percentage is 18250% but it is more than hunderd how can we statisfy it to the students. 9% means nineth part of 100 but can we say 18250% is 18250 part of hundred if yes then how. thanks for your quick response, a learner from asia.

  10. Well, remember that a percentage is always a percent of something. So if I have $9 in my wallet and you have $100 in yours, that means that my money is only 9% of your money. But that also means that your money is slightly more that 1111% of what is in my wallet.

    It is all a matter of relationship, of which thing we are treating as the “whole thing” in each comparison. I cannot have more than 100% of my own money, because my money is all that I have — but when other people compare their money to mine, they might have more. They might have twice as much money as me (200% of what is in my wallet), or they might have ten times as much (1000%).

    And I am sure that Bill Gates has much more than 18,250% of both of our wallets put together!

  11. hi
    please solve this question
    Divide Rs. 450 in such a way that thrice of the first share equals twice of the second share while four times of the second share equals thrice of the third.

  12. hi noor —

    I’m not going to give a full solution here, in case this is someone’s homework, in which case they should do the final figuring out. But I will get you started.

    This is a good problem for Singapore-style bar diagrams”.

    You would start like this:

    |—–| first share

    |—–|—–|—–| thrice first share

    |——–|——–| twice second share

    |——–| second share

    now you could subdivide the first and second share so that you can see them as being made up of portions of the same size

    |–|–| first share (2 parts)

    |–|–|–| second share (3 parts)

    Now do the same thing with 4x the second share, and thrice the third. Luckily the same divisions we just made here keep working out nicely.

    Then when you have a bar-diagram of the third share that has parts the same size as the parts of the first and second share, count up how many equal “parts” there are in total between the three shares. These have to add up to 450, so divide 450 by the number of parts, so you know how big a part is, then since the first share is 2 parts, it will be 2x the size of a part, etc.

  13. I have a question lets say that I have scored a 62.5 on test 1, test 2 66 test 3 77 and all test are worth 19% of my grade what is my percentage so far for the class and I forgot to mention that final is worth 28 % what would be my final grade if I score 77 as well on final help me please

  14. Your question is not clearly stated, Shawntee. It is easy to average your 3 tests so far, but I cannot tell if you mean they are each 19% or that their average is 19% of the grade. Either way, your percentages don’t add up to 100%.

    Here is the basic way to handle weighted scores: Multiply each score times its percentage value (the percent it is worth in your final grade). Add all these products up to make 100% of your grade.

    \left( score \times value \right) + \left( score \times value \right) + ... = final \; grade

  15. Denise,

    Maybe you are already aware of this book, but I thought it was an interesting contrast to the order in which rational number topics are usually taught. Here is an approach that starts with percents, then moves to decimals, then fractions.

    The students actually find percents easy to work with and end up thinking in terms of percents when working with rational numbers.


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