A percentage problem

Can you help me solve this question?

Jayne is given £5 for her birthday.

She spends 30% of it.

How much of her birthday money does she spend?

And... Is my method correct for this question?

100-30=70

70/5=14

1.40

No, the asker's method is not correct. In fact, it looks to me like she is randomly doing operations with numbers...

To find 30% of something, one MENTAL MATH method that I like is to first find 10% of that something.

Now, 10% of something is of course 1/10 of it. And to find 1/10 of it, just divide that by 10.

So, we divide £5 by 10 to find 1/10 of £5. It is £0.50. That's 10% of the total. And 30% of the total is three times as much, or £1.50.

So Jayne spent £1.50.

See also this video that explains how to find percentages using mental math:

A percentage problem

Can you help me solve this question?

Jayne is given £5 for her birthday.

She spends 30% of it.

How much of her birthday money does she spend?

And... Is my method correct for this question?

100-30=70

70/5=14

1.40

No, the asker's method is not correct. In fact, it looks to me like she is randomly doing operations with numbers...

To find 30% of something, one MENTAL MATH method that I like is to first find 10% of that something.

Now, 10% of something is of course 1/10 of it. And to find 1/10 of it, just divide that by 10.

So, we divide £5 by 10 to find 1/10 of £5. It is £0.50. That's 10% of the total. And 30% of the total is three times as much, or £1.50.

So Jayne spent £1.50.

See also this video that explains how to find percentages using mental math:

Math Mammoth Percent

Math Mammoth Percent book is here! Part of the "Math Mammoth Blue Series", this is a topical worktext, which contains both the instruction and the exercises in the same book. Download price: $3.50.

I know that the concept of percent and related applications are often difficult for students. I wrote this text keeping this in mind, and I hope it can clear the confusion on percentages, percent of change, and such topics!

From the description:

Math Mammoth Percent teaches students the concept of percent, percentage of a number, discounts, sales tax, percent of change, circle graphs, and percent of comparison. It is suitable for 6th-8th grade (middle school).

The concept of percent builds on a student's understanding of fractions and decimals. Specifically, students should already be very familiar with the idea of finding a fractional part of a whole (such as finding 3/4 of $240). Students who have used Math Mammoth have been practicing that concept since 4th grade, and one reason why I have emphasized finding a fractional part of a whole so much in the division and fraction materials in the earlier grades is specifically to lay a groundwork for the concept of percent.

Assuming the student has mastered how to find a fractional part of a whole, and can easily convert fractions to decimals, studying percents in middle school should not be difficult.

Read more and see the free sample pages!

Math Mammoth Percent

Math Mammoth Percent book is here! Part of the "Math Mammoth Blue Series", this is a topical worktext, which contains both the instruction and the exercises in the same book. Download price: $3.50.

I know that the concept of percent and related applications are often difficult for students. I wrote this text keeping this in mind, and I hope it can clear the confusion on percentages, percent of change, and such topics!

From the description:

Math Mammoth Percent teaches students the concept of percent, percentage of a number, discounts, sales tax, percent of change, circle graphs, and percent of comparison. It is suitable for 6th-8th grade (middle school).

The concept of percent builds on a student's understanding of fractions and decimals. Specifically, students should already be very familiar with the idea of finding a fractional part of a whole (such as finding 3/4 of $240). Students who have used Math Mammoth have been practicing that concept since 4th grade, and one reason why I have emphasized finding a fractional part of a whole so much in the division and fraction materials in the earlier grades is specifically to lay a groundwork for the concept of percent.

Assuming the student has mastered how to find a fractional part of a whole, and can easily convert fractions to decimals, studying percents in middle school should not be difficult.

Read more and see the free sample pages!

Percentages with mental math

(This is an older post that I have revised plus added a video to it.)

In this article I want to explore some ideas for using MENTAL math in calculating percents or percentages.

I have made this video of percent & mental math strategies as well:


And here are the ideas:

1. Find 10% of some example numbers (by dividing by 10).
   
   
2. Find 1% of some example numbers (by dividing by 100).
   
   
3. 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.
   
   
4. Find 3%, 4%, 6% etc. of these numbers.
   FIRST find 1% of the number, then multiply.
   
   
5. Find 15% of some numbers.
   First find 10%, halve that to find 5%, and add the two results.
   
   
6. Find 25% and then 75% of some numbers. 25% of a number is 1/4 of it, so you find it by dividing by 4. For example, 25% of 16 is 4. To find 75%, first find 25% and multiply that by 3.
   
   
7. Calculate some simple discounts. If an item is discounted 20%, 15%, 25%, 75% etc., then find the new price.
   
   For example, a book costs $40 and is discounted by 15%. What is the new price?
   First find 15% of $40 (10% of $40 is $4 and 5% of $40 is $2... so 15% of it is $6). Then subtract $40 - $6. So the new price is $34.
   
   
   
8. "40% of a number is 56. What is the number?" - types of problems.
   
   You CAN do this mentally: First FIND 10% of the number, and then multiply that result by 10, and you'll get 100% of the number - which is the number itself.
   
   So if 40% is 56, then 10% is 14 (divide by 4). Then, 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.
   
   
9. "34% of a number is 129. What is the number?" (A calculator will help here.)
   
   You don't need to write an equation. You can just 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.
   

Hope you enjoyed these little mental math ideas! They also help students understand the concept of percent where they don't end up relying too much on mechanical calculations or equations.

Percentages with mental math

(This is an older post that I have revised plus added a video to it.)

In this article I want to explore some ideas for using MENTAL math in calculating percents or percentages.

I have made this video of percent & mental math strategies as well:


And here are the ideas:

1. Find 10% of some example numbers (by dividing by 10).
   
   
2. Find 1% of some example numbers (by dividing by 100).
   
   
3. 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.
   
   
4. Find 3%, 4%, 6% etc. of these numbers.
   FIRST find 1% of the number, then multiply.
   
   
5. Find 15% of some numbers.
   First find 10%, halve that to find 5%, and add the two results.
   
   
6. Find 25% and then 75% of some numbers. 25% of a number is 1/4 of it, so you find it by dividing by 4. For example, 25% of 16 is 4. To find 75%, first find 25% and multiply that by 3.
   
   
7. Calculate some simple discounts. If an item is discounted 20%, 15%, 25%, 75% etc., then find the new price.
   
   For example, a book costs $40 and is discounted by 15%. What is the new price?
   First find 15% of $40 (10% of $40 is $4 and 5% of $40 is $2... so 15% of it is $6). Then subtract $40 - $6. So the new price is $34.
   
   
   
8. "40% of a number is 56. What is the number?" - types of problems.
   
   You CAN do this mentally: First FIND 10% of the number, and then multiply that result by 10, and you'll get 100% of the number - which is the number itself.
   
   So if 40% is 56, then 10% is 14 (divide by 4). Then, 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.
   
   
9. "34% of a number is 129. What is the number?" (A calculator will help here.)
   
   You don't need to write an equation. You can just 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.
   

Hope you enjoyed these little mental math ideas! They also help students understand the concept of percent where they don't end up relying too much on mechanical calculations or equations.

Credit card math

Considering the crisis taking place within U.S. financial institutions, it's a good reminder of how we MUST teach our kids how to work out the math with credit cards, mortgages, and any type of loans.

Murray Bourne from squareCircleZ has taken this matter to heart and has written a good lesson about math that goes into taking a loan. He's also analyzed several misleading credit card ads and figured out the TRUE interest rate (which you find out after reading the fine print).

First go here:

Credit Cards - a simplified discussion on how banks make money on credit cards. This is a must read for all people (young or old) who even consider taking a credit.

Then check these posts and warn your youngsters:

Misleading Credit Card Advertising. This ad advertises "a attractive interest rate of 5% p.a." (p.a. means per annum or yearly interest rate), but has a "small" administration fee of 6%.

And then check another misleading credit card ad which advertises a 0% p.a.*. Notice it has an asterisk... and the info in the asterisk can lead up to 24% annual interest!

Credit card math

Considering the crisis taking place within U.S. financial institutions, it's a good reminder of how we MUST teach our kids how to work out the math with credit cards, mortgages, and any type of loans.

Murray Bourne from squareCircleZ has taken this matter to heart and has written a good lesson about math that goes into taking a loan. He's also analyzed several misleading credit card ads and figured out the TRUE interest rate (which you find out after reading the fine print).

First go here:

Credit Cards - a simplified discussion on how banks make money on credit cards. This is a must read for all people (young or old) who even consider taking a credit.

Then check these posts and warn your youngsters:

Misleading Credit Card Advertising. This ad advertises "a attractive interest rate of 5% p.a." (p.a. means per annum or yearly interest rate), but has a "small" administration fee of 6%.

And then check another misleading credit card ad which advertises a 0% p.a.*. Notice it has an asterisk... and the info in the asterisk can lead up to 24% annual interest!

Percent of change

When some quantity changes, such as a price or the amount of students, we can measure either the absolute change ("The price increased by $5" or "There were 93 less students this year"), or the percent change.

In percent change, we express WHAT PART of the original quantity the change was.

For example, if a gadget costs $44 and the price is increased by $5, we measure the percent change by first considering WHAT PART $5 is of $44. Of course the answer is easy: it is 5/44 or five forty-fourths parts.

To make it percent change, however, we need to express that part using hundredths and not 44th parts. this happens to be easy, too. As seen in my previous post, you COULD make a proportion to find out how many hundredths 5/44 is:

5/44 = x/100

To solve this, you simply go 5/44 x 100, which is easy enough to remember in itself. In fact, this is the rule often given: you compare the PART to the WHOLE using division (5/44), and multiply that by 100.

---

There were 568 students one year, and 480 the next year. By how may percent did the student population decrease?

You first calculate the absolute change, which is 568 − 480 = 88. Then we find what part of the original population is 88 (it is 88/568), and express that using hundredth parts (percents):

88/568 x 100 = 15.49%

The student population decreased by 15.49%.

---

Often we are given the opposite problem: we know the percent of change and the original situation, and are asked about the new situation.

The price was $4.55 and increased by 14.78%. What is the new price?

Here, we'd need to find the price increase, or the absolute change in price first. We know the percent part of the total (it is 14.78/100) and the total amount, so multiplying those we get the part as a dollar-amount: 14.78/100 x $4.55 = $0.67249. So this is the increase. To find the new price, add the increase to the original: $4.55 + $0.67249 = $5.22249 = $5.22.

Instead of multiplying by 14.78/100, it is far quicker to multiply by 0.1478 — or to change the percent-amount 14.78% to a decimal 0.1478 and multiply by it.

And, since in the end we need to add the original total, the whole calculation looks like this:

0.1478 × $4.55 + $4.55

Here, using distributive property we can make it look like this:

= $4.55 (0.1478 + 1) = $4.55(1.1478) = 1.1478 × $4.55

So it can all be done in one multiplication. Instead of multiplying by the decimal 0.1478, you add 1 to it before multiplying.

---

Then one more possible problem type is that you know the percent of change and the actual change amount (absolute change), and are asked the original and/or the new total.

The price increased by 13%, or by $10.14. What was the original price?

Let the original price be p. Then you can build an equation based on the idea that the price increase is 13% of p:

0.13p = $10.14   or   13/100p = $10.14

p = $10.14/0.13 = $78.

The new price would be found by adding.

---

Some other lessons to read are below:

Percent Of Change - Lesson and Problems

General Increase and Decrease Examples from Purplemath.com

Percent of change calculator - enter the original and the changed quantities, and it calculates the percent of change.

PERCENT INCREASE OR DECREASE lesson from TheMathPage.com

Percent of change

When some quantity changes, such as a price or the amount of students, we can measure either the absolute change ("The price increased by $5" or "There were 93 less students this year"), or the percent change.

In percent change, we express WHAT PART of the original quantity the change was.

For example, if a gadget costs $44 and the price is increased by $5, we measure the percent change by first considering WHAT PART $5 is of $44. Of course the answer is easy: it is 5/44 or five forty-fourths parts.

To make it percent change, however, we need to express that part using hundredths and not 44th parts. this happens to be easy, too. As seen in my previous post, you COULD make a proportion to find out how many hundredths 5/44 is:

5/44 = x/100

To solve this, you simply go 5/44 x 100, which is easy enough to remember in itself. In fact, this is the rule often given: you compare the PART to the WHOLE using division (5/44), and multiply that by 100.

---

There were 568 students one year, and 480 the next year. By how may percent did the student population decrease?

You first calculate the absolute change, which is 568 − 480 = 88. Then we find what part of the original population is 88 (it is 88/568), and express that using hundredth parts (percents):

88/568 x 100 = 15.49%

The student population decreased by 15.49%.

---

Often we are given the opposite problem: we know the percent of change and the original situation, and are asked about the new situation.

The price was $4.55 and increased by 14.78%. What is the new price?

Here, we'd need to find the price increase, or the absolute change in price first. We know the percent part of the total (it is 14.78/100) and the total amount, so multiplying those we get the part as a dollar-amount: 14.78/100 x $4.55 = $0.67249. So this is the increase. To find the new price, add the increase to the original: $4.55 + $0.67249 = $5.22249 = $5.22.

Instead of multiplying by 14.78/100, it is far quicker to multiply by 0.1478 — or to change the percent-amount 14.78% to a decimal 0.1478 and multiply by it.

And, since in the end we need to add the original total, the whole calculation looks like this:

0.1478 × $4.55 + $4.55

Here, using distributive property we can make it look like this:

= $4.55 (0.1478 + 1) = $4.55(1.1478) = 1.1478 × $4.55

So it can all be done in one multiplication. Instead of multiplying by the decimal 0.1478, you add 1 to it before multiplying.

---

Then one more possible problem type is that you know the percent of change and the actual change amount (absolute change), and are asked the original and/or the new total.

The price increased by 13%, or by $10.14. What was the original price?

Let the original price be p. Then you can build an equation based on the idea that the price increase is 13% of p:

0.13p = $10.14   or   13/100p = $10.14

p = $10.14/0.13 = $78.

The new price would be found by adding.

---

Some other lessons to read are below:

Percent Of Change - Lesson and Problems

General Increase and Decrease Examples from Purplemath.com

Percent of change calculator - enter the original and the changed quantities, and it calculates the percent of change.

PERCENT INCREASE OR DECREASE lesson from TheMathPage.com

Some percent basics

The word "percent" means "per hundred", as if dividing by hundred — a hundredth part of something.

We treat some quantity (say 65 or $489 or 1.392 or anything) as "one whole". This "one whole" is then divided to hundred equal parts in our minds, and each such part is one percent of the whole.

If the "one whole" is 650 people, then 1% of it would be 6.5 people (if you have a practical application, you'd need to round such an answer to whole peoples of course).

If the "one whole" is $42, then 1% of it is $0.42. Also, 2% of it would be $0.84.

So to find 1% of something, divide by 100.
To find 24% or 8% or any other percentage, you can technically first find the 1%, then take that times 24 or 8 or whatever is your percentage.

For example:
To find 7% of $41.50, first go $41.50/100 to find 1% or 1/100 of $41.50, then multiply that by 7. But this is the same as (7/100) x $41.50, and 7/100 is 0.07 as a decimal. In most calculations, it is more practical to use decimals instead of this "divide by 100, then multiply" stuff.

So to find 7% of $41.50, I simply calculate 0.07 x $41.50 with a calculator.

To find 10% of something, you could first divide by 100 and then multiply by 10, but it's far quicker to simply divide by 10.

For example:
10% of 90 is 90/10 = 9.
10% of 250.6 is 25.06.

When you can find 10%, it's so easy to find 20%, 30%, 40%, etc., and 5% of anything just by using the 10% as a starting point.

For example:
20% of 52. First find 10% of 52 as 5.2, then double that: it is 10.4.

Example:
A gadget costs $48 and is discounted by 15%. What is the new price?

Imagine the price $48 is divided to 100 equal parts. Then you take a way 15 of those parts. That leaves 85 of those parts - what is the dollar amount that is left? Remember you're not taking away $15 but 15% of the total.

The student needs to realize that $48 is 100% - a "one whole", and 15 of those 100 parts will be taken away.

Solution:
10% of $48 would be $4.80.
5% of $48 would be $2.40 (half of 10%).
So 15% of $48 is $7.20. Subtract that from the orignal price to get the discount price of $40.80.
With a calculator, I'd go 0.85 x $48. (MAKE SURE YOU FIGURE OUT WHERE THE 0.85 COMES FROM!)

How many percent is it?

Of the class of 34 students, 12 are girls. How many percent of the class are girls?

Here, the "one whole" is 34, the whole class. The problem is, if that 34 "one whole" was divided to 100 parts, how many of those parts would we need to make 12 students?

Or, you could compare 34 people side-by-side with 100 "something". Imagine all those 34 people put head-to-toe so they form a long line, and those 12 girls are at the one end of that line. If you'd find 100 equal-size measuring units that would total exactly the same length as your people-line, how many of those measuring units would the 12 girls equal?

This easily leads to a percent proportion:

12/34 = x / 100

Solving x, you'd get
x = (12/34) × 100.

After you do this kind of proportion a few times, you notice that each time we just compare the part to the total using division, such as 12/34 in my example. So it's quite fast then to just write that directly, when solving "what percent" problems.

For example:
A $199 guitar was discounted by $40. How many percent discount was that?

Here, the "one whole" is the original (total) price, $199. It's simply asking how many percent is 40 of 199? Just calculate 40/199, and multiply the given decimal by 100 (which is easy to do mentally).

Hope this helps some. We'll tackle the percent of change next time.

Some percent basics

The word "percent" means "per hundred", as if dividing by hundred — a hundredth part of something.

We treat some quantity (say 65 or $489 or 1.392 or anything) as "one whole". This "one whole" is then divided to hundred equal parts in our minds, and each such part is one percent of the whole.

If the "one whole" is 650 people, then 1% of it would be 6.5 people (if you have a practical application, you'd need to round such an answer to whole peoples of course).

If the "one whole" is $42, then 1% of it is $0.42. Also, 2% of it would be $0.84.

So to find 1% of something, divide by 100.
To find 24% or 8% or any other percentage, you can technically first find the 1%, then take that times 24 or 8 or whatever is your percentage.

For example:
To find 7% of $41.50, first go $41.50/100 to find 1% or 1/100 of $41.50, then multiply that by 7. But this is the same as (7/100) x $41.50, and 7/100 is 0.07 as a decimal. In most calculations, it is more practical to use decimals instead of this "divide by 100, then multiply" stuff.

So to find 7% of $41.50, I simply calculate 0.07 x $41.50 with a calculator.

To find 10% of something, you could first divide by 100 and then multiply by 10, but it's far quicker to simply divide by 10.

For example:
10% of 90 is 90/10 = 9.
10% of 250.6 is 25.06.

When you can find 10%, it's so easy to find 20%, 30%, 40%, etc., and 5% of anything just by using the 10% as a starting point.

For example:
20% of 52. First find 10% of 52 as 5.2, then double that: it is 10.4.

Example:
A gadget costs $48 and is discounted by 15%. What is the new price?

Imagine the price $48 is divided to 100 equal parts. Then you take a way 15 of those parts. That leaves 85 of those parts - what is the dollar amount that is left? Remember you're not taking away $15 but 15% of the total.

The student needs to realize that $48 is 100% - a "one whole", and 15 of those 100 parts will be taken away.

Solution:
10% of $48 would be $4.80.
5% of $48 would be $2.40 (half of 10%).
So 15% of $48 is $7.20. Subtract that from the orignal price to get the discount price of $40.80.
With a calculator, I'd go 0.85 x $48. (MAKE SURE YOU FIGURE OUT WHERE THE 0.85 COMES FROM!)

How many percent is it?

Of the class of 34 students, 12 are girls. How many percent of the class are girls?

Here, the "one whole" is 34, the whole class. The problem is, if that 34 "one whole" was divided to 100 parts, how many of those parts would we need to make 12 students?

Or, you could compare 34 people side-by-side with 100 "something". Imagine all those 34 people put head-to-toe so they form a long line, and those 12 girls are at the one end of that line. If you'd find 100 equal-size measuring units that would total exactly the same length as your people-line, how many of those measuring units would the 12 girls equal?

This easily leads to a percent proportion:

12/34 = x / 100

Solving x, you'd get
x = (12/34) × 100.

After you do this kind of proportion a few times, you notice that each time we just compare the part to the total using division, such as 12/34 in my example. So it's quite fast then to just write that directly, when solving "what percent" problems.

For example:
A $199 guitar was discounted by $40. How many percent discount was that?

Here, the "one whole" is the original (total) price, $199. It's simply asking how many percent is 40 of 199? Just calculate 40/199, and multiply the given decimal by 100 (which is easy to do mentally).

Hope this helps some. We'll tackle the percent of change next time.

So many percent more

Updated with an answer... see below

I'm continuing to catch up after vacation, and spotted a good discussion about problems with percent, at MathNotations. (via Let's Play Math blog).

Here's a problem to solve, first of all:

There are 20% more girls than boys in the senior class.
What percent of the seniors are girls?

The answer is NOT that 40% are boys and 60% are girls...

You see, let's say there were 40 boys and 60 girls, 100 students total. If there are 40 boys, then 20% more than that would be 40 + 4 + 4 = 48 girls and not 60!

Try solve it. I'll let you think a little before answering it myself. Don't just rush over to the Mathnotations blog either! Use your thinking caps! I've already given you a big hint!

Update:
You can easily solve this problem by taking any example number for the number of boys. Like I did above, if you have 40 boys, you'd need 48 girls and there'd be 88 students total. What percent of the seniors are girls then? It'd be 48/88 = 0.545454... ≈ 54.55%. And 45.45% are boys.

But the same works even if you choose that there'd be 10 boys, which then means that there are 12 girls, and then the percent of girls in the class is 12/22 * 100% = 54.55%.

NOTE that this problem includes two DIFFERENT "wholes". First of all, it says "There are 20% more girls than boys in the senior class." This is a comparison, and the total number of boys is the "one whole" or the "100%". The number of girls is 20% more, or 120% of the boys.

In algebra terms, if there are p boys, then there will be 1.2p girls.

Then the final question involves a totally different "one whole" or 100%: it asks how many percent of the seniors are girls.

So the group of seniors becomes the 100% or the "whole", and all percent calculations are based on that. Therefore one will then compare the number of girls to the total number of seniors.

In algebra terms, the final answer as a decimal is 1.2p/(p + 1.2p) = 1.2/2.2

See also Denise's post about searching for the 100% in percent problems.

So many percent more

Updated with an answer... see below

I'm continuing to catch up after vacation, and spotted a good discussion about problems with percent, at MathNotations. (via Let's Play Math blog).

Here's a problem to solve, first of all:

There are 20% more girls than boys in the senior class.
What percent of the seniors are girls?

The answer is NOT that 40% are boys and 60% are girls...

You see, let's say there were 40 boys and 60 girls, 100 students total. If there are 40 boys, then 20% more than that would be 40 + 4 + 4 = 48 girls and not 60!

Try solve it. I'll let you think a little before answering it myself. Don't just rush over to the Mathnotations blog either! Use your thinking caps! I've already given you a big hint!

Update:
You can easily solve this problem by taking any example number for the number of boys. Like I did above, if you have 40 boys, you'd need 48 girls and there'd be 88 students total. What percent of the seniors are girls then? It'd be 48/88 = 0.545454... ≈ 54.55%. And 45.45% are boys.

But the same works even if you choose that there'd be 10 boys, which then means that there are 12 girls, and then the percent of girls in the class is 12/22 * 100% = 54.55%.

NOTE that this problem includes two DIFFERENT "wholes". First of all, it says "There are 20% more girls than boys in the senior class." This is a comparison, and the total number of boys is the "one whole" or the "100%". The number of girls is 20% more, or 120% of the boys.

In algebra terms, if there are p boys, then there will be 1.2p girls.

Then the final question involves a totally different "one whole" or 100%: it asks how many percent of the seniors are girls.

So the group of seniors becomes the 100% or the "whole", and all percent calculations are based on that. Therefore one will then compare the number of girls to the total number of seniors.

In algebra terms, the final answer as a decimal is 1.2p/(p + 1.2p) = 1.2/2.2

See also Denise's post about searching for the 100% in percent problems.

Math Mammoth Grade 7 (prealgebra) worksheets available

Just as of today, I got online and available the new Math Mammoth Grade 7-A and 7-B worksheets collections.

These pre-algebra sheets contain problems for beginning algebra topics such as expressions, linear equations, and slope - plus your typical 7th grade math topics such as integers, fractions, decimals, geometry, statistics, and probability.

You will find the problems are very varying as these worksheets have been created one by one (not script made).

Go download and enjoy the free sample sheets! - including a Percent Fact Sheet.

Math Mammoth Grade 7 (prealgebra) worksheets available

Just as of today, I got online and available the new Math Mammoth Grade 7-A and 7-B worksheets collections.

These pre-algebra sheets contain problems for beginning algebra topics such as expressions, linear equations, and slope - plus your typical 7th grade math topics such as integers, fractions, decimals, geometry, statistics, and probability.

You will find the problems are very varying as these worksheets have been created one by one (not script made).

Go download and enjoy the free sample sheets! - including a Percent Fact Sheet.

Calculating percent with mental math

Would you say that students' understanding of percent is sometimes - or often - hazy?

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.

1. Find 10% of some example numbers (by dividing by 10).
   
   
2. Find 1% of some example numbers (by dividing by 100).
   
   
3. 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.
   
   
4. Find 3%, 4%, 6% etc. of these numbers.
   FIRST find 1% of the number, then multiply.
   
   
5. Find 15% of some numbers.
   First find 10%, halve that to find 5%, and add the two results.
   
   
6. Calculate some simple discounts. If an item is discounted 20%, 15%, etc., then find the new price.
   
   
7. "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.
   
   
8. "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.)

Calculating percent with mental math

Would you say that students' understanding of percent is sometimes - or often - hazy?

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.

1. Find 10% of some example numbers (by dividing by 10).
   
   
2. Find 1% of some example numbers (by dividing by 100).
   
   
3. 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.
   
   
4. Find 3%, 4%, 6% etc. of these numbers.
   FIRST find 1% of the number, then multiply.
   
   
5. Find 15% of some numbers.
   First find 10%, halve that to find 5%, and add the two results.
   
   
6. Calculate some simple discounts. If an item is discounted 20%, 15%, etc., then find the new price.
   
   
7. "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.
   
   
8. "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.)