# Little math tricks for business and pleasure

June 29, 2021 3:07 AM Subscribe

I am pretty terrible at maths, nonetheless I enjoy D&D and have work tasks that involve discounts, conversions, etc. I am especially interested in percentage tricks and also calculator workflows.

You probably know better than I do what typical tasks might be, but things like this are what I have in mind. I also lack language to describe these things.

I learned that you can multiply a number by .28 to get what 28% of that number is.

Dividing that number by .28 would tell you what the full value is if that number is 28% of it.

How do I do if I want to increase a number by X%, let's say I have 10, and I want to increase it by 5%, I know myself that that's 10.5, but I know it by doing the two parts of the sum. Are there easier ways? On a calculator I could do 10+5 (hit % button) then =, but in person I have to first do 10x.05 and then remember that and add back the ten, etc.

Another one I find difficult is "how many percent more or less is this than that". So how much has my time increased this week compared to last. Ten hours last week and 12 hours this week is a ... and that's where my brain freezes, 2 as a percentage of ten, that's a fifth, but I know that cos I know it. So I worked 120% of the time I worked last week. An increase of 20%, is that the right way to describe it? I feel like such an idiot.

It feels there are also known "traps" I could fall into and think I have the right answer but not have. Any advice on avoiding that? Other day-to-day maths tricks?

You probably know better than I do what typical tasks might be, but things like this are what I have in mind. I also lack language to describe these things.

I learned that you can multiply a number by .28 to get what 28% of that number is.

Dividing that number by .28 would tell you what the full value is if that number is 28% of it.

How do I do if I want to increase a number by X%, let's say I have 10, and I want to increase it by 5%, I know myself that that's 10.5, but I know it by doing the two parts of the sum. Are there easier ways? On a calculator I could do 10+5 (hit % button) then =, but in person I have to first do 10x.05 and then remember that and add back the ten, etc.

Another one I find difficult is "how many percent more or less is this than that". So how much has my time increased this week compared to last. Ten hours last week and 12 hours this week is a ... and that's where my brain freezes, 2 as a percentage of ten, that's a fifth, but I know that cos I know it. So I worked 120% of the time I worked last week. An increase of 20%, is that the right way to describe it? I feel like such an idiot.

It feels there are also known "traps" I could fall into and think I have the right answer but not have. Any advice on avoiding that? Other day-to-day maths tricks?

Best answer:

x% of y = y% of x

example: 17% of 50 = .17 * 50 = 8.5

50% of 17 = .5 * 17 = 8.5

this is because "x% of y" is:

(x/100) * y

or

x * (1/100) * y

and since multiplication is commutative, you can rearrange that to show that it is the same as y% of x

posted by thelonius at 4:05 AM on June 29, 2021 [5 favorites]

*percentage tricks*x% of y = y% of x

example: 17% of 50 = .17 * 50 = 8.5

50% of 17 = .5 * 17 = 8.5

this is because "x% of y" is:

(x/100) * y

or

x * (1/100) * y

and since multiplication is commutative, you can rearrange that to show that it is the same as y% of x

posted by thelonius at 4:05 AM on June 29, 2021 [5 favorites]

Best answer:

10 x 1.05

When you increase 10 by 5%, you are calculating 105% of 10. 100% of 10 is obviously just 10, the same as 10 x 1.00 is still just 10, so you can imagine the 1.00 is representing 100%.

Examples:

Increase 10 by 28%? That will result in 128% of 10, or 10 x 1.28

Increase 48 by 87%? That will result in 187% of 48, or 48 x 1.87

Increase 3 by 237%? That will result in 337%* of 3, or 3 x 3.37

*not 237% because you need to include the original 100%.

posted by EndsOfInvention at 4:27 AM on June 29, 2021 [1 favorite]

*let's say I have 10, and I want to increase it by 5%*10 x 1.05

When you increase 10 by 5%, you are calculating 105% of 10. 100% of 10 is obviously just 10, the same as 10 x 1.00 is still just 10, so you can imagine the 1.00 is representing 100%.

Examples:

Increase 10 by 28%? That will result in 128% of 10, or 10 x 1.28

Increase 48 by 87%? That will result in 187% of 48, or 48 x 1.87

Increase 3 by 237%? That will result in 337%* of 3, or 3 x 3.37

*not 237% because you need to include the original 100%.

posted by EndsOfInvention at 4:27 AM on June 29, 2021 [1 favorite]

Best answer: One thing I find helpful is approximations. Like, I don’t know what 2/7 is off the top of my head, and most of the time it’s not important enough for me to actually calculate it. But I know what 2/6 is (.33333) and I know what 2/8 is (.25), and 2/7 is between those two, so it’s probably around .3, maybe a little less like .29 or something. If I need to be more specific than that, I’d use a calculator anyway.

When I was a kid, I learned my times tables for the number 7 by watching American football. (Rugby works as well.) That’s helpful because there’s no easy way to tell if a number is divisible by 7.

Percent more or less is just the difference (x-y) divided by the first amount. If you worked 10 hours last week and 12 this week, that’s a difference of 2, and 2/10 is .2. 20% more. You can use the approximation trick for less even numbers. 19 hours last week, 16 this week, difference of 3, 3/19 is roughly the same as 3/20 which is .15. ~15% less.

posted by kevinbelt at 5:19 AM on June 29, 2021 [3 favorites]

When I was a kid, I learned my times tables for the number 7 by watching American football. (Rugby works as well.) That’s helpful because there’s no easy way to tell if a number is divisible by 7.

Percent more or less is just the difference (x-y) divided by the first amount. If you worked 10 hours last week and 12 this week, that’s a difference of 2, and 2/10 is .2. 20% more. You can use the approximation trick for less even numbers. 19 hours last week, 16 this week, difference of 3, 3/19 is roughly the same as 3/20 which is .15. ~15% less.

posted by kevinbelt at 5:19 AM on June 29, 2021 [3 favorites]

I watched a Great Courses on mental math tricks a few years ago and it was really helpful in just feeling confident about doing math in my head. It was very corny but the guy’s enthusiasm was infectious. Might be worth signing up for the free trial to watch?

Mostly it comes down to practice, though. I annoyed the crap out of my husband but now I can quickly multiply any number by 11 in my head — which is not all that useful, admittedly, but with other more useful things you’ll get more opportunities to practice.

I find it much easier to do percents by relating them to 10% — for example I know that 10% is easy to find by chopping a zero off/moving the decimal one time to the left

200 —> 20

20 —> 2

300,000 —-> 30,000

From there I can do 5%, which is just half of 10%

Or 20% which is just double

But in my state sales tax is 7.5% so I frequently find 10%, then half to get 5% then half again to get 2.5%. I add together what I got from the two “halving” and that’s 7.5%

If you want to find the price of something after a discount, do 100 minus discount% and then use the 10% trick above or use 50% as your benchmark

So 60% off means it’s 40% of the original price

MSRP: 40

I know half of $40 is 20 and I need to subtract another 10%, which is 4 - then 20-4 so pre-tax $16

With tax:

10% of 16 is 1.6; 5% is 0.8, 2.5% is 0.4 —> 0.8+0.4=1.2

Final price is $17.20

posted by shesdeadimalive at 5:54 AM on June 29, 2021 [1 favorite]

Mostly it comes down to practice, though. I annoyed the crap out of my husband but now I can quickly multiply any number by 11 in my head — which is not all that useful, admittedly, but with other more useful things you’ll get more opportunities to practice.

I find it much easier to do percents by relating them to 10% — for example I know that 10% is easy to find by chopping a zero off/moving the decimal one time to the left

200 —> 20

20 —> 2

300,000 —-> 30,000

From there I can do 5%, which is just half of 10%

Or 20% which is just double

But in my state sales tax is 7.5% so I frequently find 10%, then half to get 5% then half again to get 2.5%. I add together what I got from the two “halving” and that’s 7.5%

If you want to find the price of something after a discount, do 100 minus discount% and then use the 10% trick above or use 50% as your benchmark

So 60% off means it’s 40% of the original price

MSRP: 40

I know half of $40 is 20 and I need to subtract another 10%, which is 4 - then 20-4 so pre-tax $16

With tax:

10% of 16 is 1.6; 5% is 0.8, 2.5% is 0.4 —> 0.8+0.4=1.2

Final price is $17.20

posted by shesdeadimalive at 5:54 AM on June 29, 2021 [1 favorite]

A few fundamental rules that you perhaps have never been made explicitly aware of:

Then what you're trying to do is the same thing as finding (100 + 5)% of 10 (rule 3)

Which is the same thing as finding 105% of 10

Which is the same thing as finding 105% multiplied by 10 (rule 2)

Which is the same thing as calculating (105 ÷ 100) multiplied by 10 (rule 1)

Which is the same thing as 10.5

10 hours last week has increased by X% to 12 hours this week

Which is the same thing as saying that 12 hours is (X + 100)% of 10 hours (rule 3)

Which is the same thing as saying that 12 hours is (X + 100)% × 10 hours (rule 2)

Which is the same thing as saying that 12 hours ÷ 10 hours is (X + 100)%

Which is the same thing as saying that (X + 100)% is 1.2

Which is the same thing as saying that (X + 100) ÷ 100 is 1.2 (rule 1)

Which is the same thing as saying that X + 100 is 1.2 × 100

Which is the same thing as saying that X + 100 is 120

Which is the same thing as saying that X is 20

So 10 hours last week has increased by 20% to 12 hours this week.

Another example: Australia's Goods and Services Tax (GST) increases the price of almost everything that can be sold by 10%. If my total bill for having a small deck installed is $2000, how much of that is going to the builder and how much is going to the Australian Tax Office?

$2000 is 10%

Which is the same thing as saying that $2000 is 110%

Which is the same thing as saying that $2000 is (110 ÷ 100) × what the builder gets.

Which is the same thing as saying that $2000 is 1.1 × what the builder gets.

Which is the same thing as saying that $2000 ÷ 1.1 is what the builder gets.

Which is the same thing as saying that $1818.18 is what the builder gets, which means that the other $181.82 is going to the tax office.

And, as a reality check: $181.82 is indeed 10% of $1818.82 (i.e. 0.1 × $1818.82), to the nearest cent.

Most of gaining competence with day-to-day maths involves a willingness to take the time to sit down with paper and pen and break these things down into steps you

posted by flabdablet at 6:22 AM on June 29, 2021 [1 favorite]

- "%" is nothing more nor less than a shorthand way of writing "÷ 100"
- "of" means the same thing as "multiplied by"
- "increase Y by X%" means the same thing as "find (100 + X)% of Y"
- "decrease Y by X%" means the same thing as "find (100 - X)% of Y"

*let's say I have 10, and I want to increase it by 5%*Then what you're trying to do is the same thing as finding (100 + 5)% of 10 (rule 3)

Which is the same thing as finding 105% of 10

Which is the same thing as finding 105% multiplied by 10 (rule 2)

Which is the same thing as calculating (105 ÷ 100) multiplied by 10 (rule 1)

Which is the same thing as 10.5

*So how much has my time increased this week compared to last. Ten hours last week and 12 hours this week is*10 hours last week has increased by X% to 12 hours this week

Which is the same thing as saying that 12 hours is (X + 100)% of 10 hours (rule 3)

Which is the same thing as saying that 12 hours is (X + 100)% × 10 hours (rule 2)

Which is the same thing as saying that 12 hours ÷ 10 hours is (X + 100)%

Which is the same thing as saying that (X + 100)% is 1.2

Which is the same thing as saying that (X + 100) ÷ 100 is 1.2 (rule 1)

Which is the same thing as saying that X + 100 is 1.2 × 100

Which is the same thing as saying that X + 100 is 120

Which is the same thing as saying that X is 20

So 10 hours last week has increased by 20% to 12 hours this week.

Another example: Australia's Goods and Services Tax (GST) increases the price of almost everything that can be sold by 10%. If my total bill for having a small deck installed is $2000, how much of that is going to the builder and how much is going to the Australian Tax Office?

$2000 is 10%

*more than*what the builder gets.Which is the same thing as saying that $2000 is 110%

*of*what the builder gets.Which is the same thing as saying that $2000 is (110 ÷ 100) × what the builder gets.

Which is the same thing as saying that $2000 is 1.1 × what the builder gets.

Which is the same thing as saying that $2000 ÷ 1.1 is what the builder gets.

Which is the same thing as saying that $1818.18 is what the builder gets, which means that the other $181.82 is going to the tax office.

And, as a reality check: $181.82 is indeed 10% of $1818.82 (i.e. 0.1 × $1818.82), to the nearest cent.

Most of gaining competence with day-to-day maths involves a willingness to take the time to sit down with paper and pen and break these things down into steps you

*do*understand, then doing them step by step. Blindly applying shortcuts*that you haven't worked out for yourself*leads you astray with day-to-day maths about as often as for anything else, because the shortcuts get shoehorned onto cases they don't really fit. There is really no substitute for a genuine understanding of what we're doing.*First*get it right.*Then*practise until it's easy.*Then*practise more until it's fast. That's really all that those people who are better than you at*anything*have been doing while you weren't watching them.posted by flabdablet at 6:22 AM on June 29, 2021 [1 favorite]

Best answer:

Others have given you the easier ways, but there is nothing wrong with doing it in two parts, if by that you mean computing 5% and then adding that to the original number. If I have to do this math in my head, that's exactly what I'll do.

There is an advantage to using a process that you fully understand compared to one that might be faster, but seems like magic to you.

posted by It's Never Lurgi at 7:21 AM on June 29, 2021 [3 favorites]

*How do I do if I want to increase a number by X%, let's say I have 10, and I want to increase it by 5%, I know myself that that's 10.5, but I know it by doing the two parts of the sum. Are there easier ways?*Others have given you the easier ways, but there is nothing wrong with doing it in two parts, if by that you mean computing 5% and then adding that to the original number. If I have to do this math in my head, that's exactly what I'll do.

There is an advantage to using a process that you fully understand compared to one that might be faster, but seems like magic to you.

posted by It's Never Lurgi at 7:21 AM on June 29, 2021 [3 favorites]

Response by poster: These are amazing, I'll be back to best-answer more later when I can absorb them as I recognise the information in them is useful to, as flabdablet says, "get it right", but I have already marked the ones where I am like "

I don't have tax or tipping based stuff that I do, but I realised I do a bit of the same sort of maths at the gym, combining weights, 2x2.5+2x5+15 for example, and then when I want to go up 5 total from there, what do I need to switch out.

posted by J.R. Hartley at 8:20 AM on June 29, 2021 [1 favorite]

*bam*that's cheating, yay!". I'm loving these!I don't have tax or tipping based stuff that I do, but I realised I do a bit of the same sort of maths at the gym, combining weights, 2x2.5+2x5+15 for example, and then when I want to go up 5 total from there, what do I need to switch out.

posted by J.R. Hartley at 8:20 AM on June 29, 2021 [1 favorite]

My only percentage based trick is that 10% of any number is just moving the decimal place to the left one place. Every other number can be made of 10 — half is 5, half of 5 is 2.5, half of 2.5 is 1.25; going up 20% is twice 10, etc., so that means you can just add those values together to cobble together the number you want. (If you _were_ tipping, say 20%, then it's just move the decimal and double, for instance. If your tax was 7.5% that's 5 + half five.)

I also go back and forth between fractions and percentages (decimals) as it's easy. In two fractions, the cross multiplications are equal if the fractions are equal so I use that a lot, too. If x/y and m/n are equal xn = my, so when you need to solve for x, you can do x = my / n. I don't know why but I seem to need to preserve ratios a lot.

posted by dame at 8:57 AM on June 29, 2021

I also go back and forth between fractions and percentages (decimals) as it's easy. In two fractions, the cross multiplications are equal if the fractions are equal so I use that a lot, too. If x/y and m/n are equal xn = my, so when you need to solve for x, you can do x = my / n. I don't know why but I seem to need to preserve ratios a lot.

posted by dame at 8:57 AM on June 29, 2021

If you want to get stuck right in to rules of thumb that work for non-obvious reasons you might enjoy the Trachtenberg Speed System of Basic Mathematics.

posted by flabdablet at 9:24 AM on June 29, 2021 [3 favorites]

posted by flabdablet at 9:24 AM on June 29, 2021 [3 favorites]

*combining weights, 2x2.5+2x5+15 for example, and then when I want to go up 5 total from there, what do I need to switch out.*

As long as what you're switching in weighs five more than what you're switching out, you're good. There will often be many ways to do that. So in this case you could switch out the two fives for another fifteen, or one of the fives for a ten if you have it, or both the 2.5s for a pair of fives.

It's essentially the same kind of problem as making change for a cash payment.

posted by flabdablet at 9:28 AM on June 29, 2021

You bought something for $125.43. Sales tax is 8.25%. How much was the price before sales tax? How much was the sales tax total?

Divide the total by 1+ the tax rate, or 1.0825 in this case. $125.43 / 1.0825 = $115.87, the price before tax. $115.87 * .0825 = $9.56. Total $115.87 and $9.56 = $125.43.

posted by Midnight Skulker at 3:39 PM on June 29, 2021

Divide the total by 1+ the tax rate, or 1.0825 in this case. $125.43 / 1.0825 = $115.87, the price before tax. $115.87 * .0825 = $9.56. Total $115.87 and $9.56 = $125.43.

posted by Midnight Skulker at 3:39 PM on June 29, 2021

Response by poster: So, to check I am grasping this, and again using money to keep it relatable; if an item is 9 dollars one place and 13 somewhere else, I can do a mental approximation that since 9 is about 75% of 12 then the higher cost is about 25% higher?

Is there a method similar to the first 3 responses in the thread to work out the actual amount?

posted by J.R. Hartley at 1:58 AM on June 30, 2021

Is there a method similar to the first 3 responses in the thread to work out the actual amount?

posted by J.R. Hartley at 1:58 AM on June 30, 2021

Best answer:

It's important to keep track of your basis for comparison.

$9 is 25% lower than $12, but $12 is a smidge over 33% higher than $9.

This is because the difference between $12 and $9 - that is, $3 - is a quarter of $12, but it's a

Another way to think about this is that for the same price you'd pay for 9 of the $13 items, you could get 13 of the $9 items.

posted by flabdablet at 3:18 AM on June 30, 2021

*since 9 is about 75% of 12 then the higher cost is about 25% higher?*It's important to keep track of your basis for comparison.

$9 is 25% lower than $12, but $12 is a smidge over 33% higher than $9.

This is because the difference between $12 and $9 - that is, $3 - is a quarter of $12, but it's a

*third*of $9.Another way to think about this is that for the same price you'd pay for 9 of the $13 items, you could get 13 of the $9 items.

posted by flabdablet at 3:18 AM on June 30, 2021

This thread is closed to new comments.

100%of it) PLUS the increase (7%). And that means you'd multiply by 1.07posted by Harvey Kilobit at 3:45 AM on June 29, 2021 [5 favorites]