Why does multiplying by 100 convert a fraction or decimal....

[Cliff Hanley]

...to a percentage?

I know that 0.5 = 5/10 = 1/2 = 50%. But I'm not sure why multiplying 0.5 and 1/2 by 100 gives us them as a percentage.

Take 1/2. Multiplying it by 100 means multiplying the numerator by 100. So 1/2 becomes 100/2 which gives 50 (50%). I can see that it works. But I don't know why. Likewise for decimals.

I have had a stab at it. To turn 1/2 into a percentage (without using the multiply by 100 method) I would look at how to turn the denominator into a 100 (multiply by 50 in this case) and then multiply the numerator by 50 also. This is effectively multiplying the whole thing by 50/50 - but 50/50 equals 1. Likewise for multiplying 1/4 by 25/25. What am I missing here?

 

[mfb]

That's just the definition of "percent" (from latin: "per hundred"). You replace "/100" by the percent sign.

 

[SteamKing]

I have had a stab at it. To turn 1/2 into a percentage (without using the multiply by 100 method) I would look at how to turn the denominator into a 100 (multiply by 50 in this case) and then multiply the numerator by 50 also. This is effectively multiplying the whole thing by 50/50 - but 50/50 equals 1. Likewise for multiplying 1/4 by 25/25. What am I missing here?

If you multiply a number by 1, you haven't changed the number.

For example, A * 1 = A

What you are doing here with $\frac{1}{2}$ is multiplying it by 1 in the form of $\frac{50}{50}$, thus

$\frac{1}{2}×\frac{50}{50} = \frac{50}{100} = 50\%$

 

[Mentallic]

This is effectively multiplying the whole thing by 50/50 - but 50/50 equals 1

Multiplying by 1 is a very simple yet powerful idea, as long as you choose the appropriate values in the numerator and denominator to multiply by for your particular problem

 

[Cliff Hanley]

That's just the definition of "percent" (from latin: "per hundred"). You replace "/100" by the percent sign.

Thanks. But I still don't understand why multiplying the fraction by 100/1 gives us the percentage. If we multiply 1/2 by 100/1 we are effectively multiplying the number 1 (the numerator) by 100. One half becomes a hundred halves. I know that 100/2 = 50/1 which gives us the correct answer (50%) but I'm not yet grasping why.

 

[Nidum]

Percent really just means fraction of 100 .

For any fraction do an operation which changes denominator to 100 . Then numerator is the percentage .

Thus 3/4 change to 75/100 ie 75%

18/6 change to 300/100 ie 300%

27/32 change to 84.3/100 ie 84.3%

 

[Cliff Hanley]

Thanks. 18/6 is an interesting one. I can see why it's 300% using the times 100/1 method. But when I first looked at it I asked myself what can I multiply 6 by to get 100 and had to get my calculator out. For the calculation 6/100 it gave an answer of 16.6666667 which is obviously 16.6 recurring but rounded up to 7 decimal places. The same for 18/300. But when you multiply (not using the calculator) the rounded up number of 16.6666667 by 6 you get 100.000002; and by 18 you get 300.000006. And when you multiply them using the calculator you get 100 and 300 respectively. The calculator clearly sees the 16.6666667 it gave as the answer as 16.6 recurring when we ask it to multiply be 6 and 18.

What hit me as profound was the notion of multiplying a number by a recurring decimal. 16.6 recurring means 16.6 followed by a (theoretically?) infinite number of 6's. How can we do this when we can never get to the end of the line of 6's to begin performing the calculation?

 

[Nidum]

Sometimes easier when doing repetitive calculations to convert fractions to decimal numbers first and then multiply by 100 .

Thus 27/32 = 0.843 . Multiply by 100 to get 84.3%

 

[HallsofIvy]

Thanks. 18/6 is an interesting one. I can see why it's 300% using the times 100/1 method. But when I first looked at it I asked myself what can I multiply 6 by to get 100 and had to get my calculator out. For the calculation 6/100 it gave an answer of 16.6666667 which is obviously 16.6 recurring but rounded up to 7 decimal places.

Then I suggest you get a new calculator! Or learn to use it correctly. 6/100= 0.006. Apparently what you are doing is 100/6, not 6/100.

The same for 18/300. But when you multiply (not using the calculator) the rounded up number of 16.6666667 by 6 you get 100.000002; and by 18 you get 300.000006. And when you multiply them using the calculator you get 100 and 300 respectively. The calculator clearly sees the 16.6666667 it gave as the answer as 16.6 recurring when we ask it to multiply be 6 and 18.

What hit me as profound was the notion of multiplying a number by a recurring decimal. 16.6 recurring means 16.6 followed by a (theoretically?) infinite number of 6's. How can we do this when we can never get to the end of the line of 6's to begin performing the calculation?

There is nothing "profound" here. "16.6 recurring" is 16 and 2/3. There is nothing "infinite" about that. Whether or not a number has a "infinite number of digits" is entirely dependent on the numeration system you are using to represent the numbers and has nothing to do with the number itself. The reason the calculator "sees" the number as you say is that it can only do a finite number of decimal places so must round off while a human can grasp the concept of recurring decimals and treat the number as a fraction.

 

[Mentallic]

You have some fraction a/b and want to know what that is as a percentage. Well, simply multiplying numerator and denominator by 100 (which doesn't change the value because you're multiplying by 1) gives us

\frac{100a}{100b}

which is equivalent to

\frac{100\frac{a}{b}}{100}

and we know that /100 means percentage, so we can then change that fraction to

100\frac{a}{b}\%

Hence whatever value a/b is in decimal, just multiply by 100 and that is your percentage.

 

[Mark44]

Thanks. 18/6 is an interesting one. I can see why it's 300% using the times 100/1 method. But when I first looked at it I asked myself what can I multiply 6 by to get 100 and had to get my calculator out. For the calculation 6/100 it gave an answer of 16.6666667 which is obviously 16.6 recurring but rounded up to 7 decimal places.

Then I suggest you get a new calculator! Or learn to use it correctly. 6/100= 0.006.

No, 6/100 = 0.06, not 0.006.

Apparently what you are doing is 100/6, not 6/100.

Yes, that has to be what he's doing.

 

[Mark44]

Thanks. But I still don't understand why multiplying the fraction by 100/1 gives us the percentage. If we multiply 1/2 by 100/1 we are effectively multiplying the number 1 (the numerator) by 100. One half becomes a hundred halves. I know that 100/2 = 50/1 which gives us the correct answer (50%) but I'm not yet grasping why.

You're making this much harder than it really is. All that's happening is converting a fraction to its decimal form, and then changing that to a percent. It's easiest if there are two digits to the right of the decimal point (making the decimal already in hundredths).

1/2 = .50, which is literally 50 hundredths. As already pointed out, "percent" means "per hundred." 50 hundredths is 50 out of one hundred or 50 per cent (50%)
1/4 = .25 or 25 hundredths, or 25 per cent (25%)
1/8 = .125 or $12 \frac 1 2$ hundredths, or or $12 \frac 1 2$%.