Calculating percent with mental math
Would you say that students' understanding of percent is sometimes - or often - hazy?
Solving that way, the unknown number would be Y, "of" would be multiplication, and "is" corresponds to '='. We'd get:
Y × 79.5% = 101.
0.795 Y = 101
Y = 101/0.795 = 127.044025157
I'm a bit leery of this method, as it's so mechanical. What if a question comes that is not worded exactly as the ones in the book, and the student just gets stuck? Or it is worded so that the student gets misled and calculates it wrong?
So while this idea is great and works, it is also necessary for students to understand the concept of percent well.
In the above problem, we are to find a number so that 79.5% of that number is 101. (Obviously, then, the number itself is more than 101.) If you understand the problem, and say the problem that way, it is pretty obvious how to write the equation:
"79.5% of that number is 101"... so 0.795 Y = 101.
Ideas for using MENTAL math for calculating percent problems
I have also made a video of this topic. It shows you how to use mental math for calculating simple percentages.
I recently got this sort of homework question sent to me:
You simply first find 10% of $50, then use that to find 30% of $50, and lastly subtract. Easy as a pie!
(10% of $50 is $5. 30% of $50 is three times as much, or $15. Lastly subtract $50 - $15 = $35. So the discounted price is #$35.)
Find the number of which 79.5% is 101.Often, solving these kinds of problems is taught with the idea that you "translate" certain words in the problem into certain symbols, and thus build an equation.
Solving that way, the unknown number would be Y, "of" would be multiplication, and "is" corresponds to '='. We'd get:
Y × 79.5% = 101.
0.795 Y = 101
Y = 101/0.795 = 127.044025157
I'm a bit leery of this method, as it's so mechanical. What if a question comes that is not worded exactly as the ones in the book, and the student just gets stuck? Or it is worded so that the student gets misled and calculates it wrong?
So while this idea is great and works, it is also necessary for students to understand the concept of percent well.
In the above problem, we are to find a number so that 79.5% of that number is 101. (Obviously, then, the number itself is more than 101.) If you understand the problem, and say the problem that way, it is pretty obvious how to write the equation:
"79.5% of that number is 101"... so 0.795 Y = 101.
Ideas for using MENTAL math for calculating percent problems
I have also made a video of this topic. It shows you how to use mental math for calculating simple percentages.
- Find 10% of some example numbers (by dividing by 10).
- Find 1% of some example numbers (by dividing by 100).
- Find 20%, 30%, 40% etc. of these numbers.
FIRST find 10% of the number, then multiply by 2, 3, 4, etc.
For example, find 20% of 18. Find 40% of $44. Find 80% of 120.
I know you can teach the student to go 0.2 × 18, 0.4 × 0.44, and 0.8 × 120 - however when using mental math, the above method seems to me to be more natural. - Find 3%, 4%, 6% etc. of these numbers.
FIRST find 1% of the number, then multiply. - Find 15% of some numbers.
First find 10%, halve that to find 5%, and add the two results. - Calculate some simple discounts. If an item is discounted 20%, 15%, etc., then find the new price.
- "40% of a number is 56. What is the number?" - types of problems.
You can do this mentally, too: First FIND 10% and then multiply that result by 10, to find 100% of the number (which is the number itself).
If 40% is 56, then 10% is 14. So 100% of the number is 140. This result is reasonable, because 40% of this number was 56, so the actual number (140) needs to be more than double that. - "34% of a number is 129. What is the number?" (Now you need a calculator.)
You don't need to write an equation. You could also first find 1% of this number, and then find 100% of the number.
If 34% of a number is 129, then 1% of that number is 129/34. Find that, and multiply the result by 100.
I recently got this sort of homework question sent to me:
I have a problem and I don't know how to solve it so here is the problem: At a popular clothing store clothes are on sale when they have hung on the rack too long. When an item is first put on sale, the store marks the prices down 30% off. If some shoes are regular-priced at $50.00, how much will they cost after the discount?
You simply first find 10% of $50, then use that to find 30% of $50, and lastly subtract. Easy as a pie!
(10% of $50 is $5. 30% of $50 is three times as much, or $15. Lastly subtract $50 - $15 = $35. So the discounted price is #$35.)
Comments
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I think, though, that you may be giving the "translation" technique a little less credit than it deserves. After all, you yourself turned the sentence around to "79.5% of that number is 101" and then used the fact that "of" signals multiplication to translate that statement into an equation. You may not think of it as a translation, because you've done so many percent problems in your life that the step is obvious to you. But if a student doesn't know how to work these problems, then it really isn't "obvious now how to write the equation"--it's just pulling a magic trick out of a hat!
On the other hand, I really like your approach to percents through mental math--a great way to build understanding!
A 20% discount of $40 is 0.8 x 40 = 32. This method is useful in writing computer code. When calculating taxes in code, it is shorter to write [cost x 1.07 = total cost.]
Example 1215 is ?% of 99875
1215/99875 = 0.012165207 which written as a percent is 1.2165207%