Math for Grownups
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In Basic Math Review

Finding Percentages and the Numbers That Go With Them

So yesterday, we reviewed some really basic stuff about percentages. Like: 10% is the same thing as 1/10 or 0.1. Easy peasy, right? Well, today it’s time to really put this stuff to work, finding percentages of numbers or the numbers, given the percentages. Oy. I can hear you groaning from here.

Most folks forget when to multiply and when to divide. So I’m going to show you a process that works no matter what kind of percentage problem you’re doing. For reals. It’s why it was important for you to know about turning percentages into fractions. Let’s start with an example.

You’ve had your eye on a gorgeous cashmere sweater for months and it’s finally on sale. But can you afford it? The original price is $125, but it’s now on sale for 30% off. Do you multiply or divide or what to find out what you’d be saving with this sale?

All you need for this problem — and pretty much all other percentage problems — is to set up a proportion. What is that, you ask? A proportion is made up of two equal ratios or fractions. The proportion you need for a percentage problem is this one:

If you can remember this proportion — and how to use it — you’re home free. So let’s dissect it a bit to help you remember. The fraction (or ratio) on the right of the proportion represents the percentage itself. You should recognize this from yesterday, when you learned to change a percent to a fraction, right? So in this problem, that ratio will be 30 over 100. That’s because the sweater is 30% off.

The ratio on the left is a little tricker, but not by much. It is the percent off of the sweater over the original price of the sweater: the part of the price over the whole price. Got it? The original price (or whole price) is $125. But we don’t know the discount (or part of the price). Let’s call that x.

DON’T PANIC! That little old x isn’t going to hurt you one bit. Promise. Just because you have an x in your math problem does not make it too challenging to solve.

But yes, you will need to solve for x. This involves two, very simple steps: Cross multiply and then get x by itself. There are tons and tons of shortcuts for this kind of a problem, but for now, we’re going to stick with the more scenic route.

To cross multiply, just multiply the x by 100 and then the 125 by the 30.

100x = 125 • 30Do you have to have the equation in that order? Nope. 125 • 30 = 100x works the same way. Heck you can even multiply in any order. Now, just start simplifying and getting x by itself:

Now, remind me, what is x? Is the price of the sweater? Nope. It’s what you would save if you bought the sweater at 30% off. The sale price of the sweater is $125 – $37.5 or $87.50.

That wasn’t so painful, was it?

But what if you needed to know what percent a number was of another number? Let’s say you just had lunch with your dad, who is known for being a bit stingy. He left a $7.50 tip on a $50 check. Was it enough? Well, set up that proportion, why don’t you?

What’s the whole? $50 or the total cost of lunch. And what’s the part? That would be the tip or $7.50. You are trying to find the percent, and 100 is always 100. Substitute, cross multiply, isolate x and voila!

Looky there, good old Dad did okay with the tip — 15%.

You can also use this proportion to find the whole, when you know the percentage and the part. Just substitute what you know, shove xin there for what you want to find and follow the same darned steps as the previous examples.

Seriously ya’ll, if you can remember this one proportion, percentages will no longer be a huge stumbling block. But I can hear a couple of you whining: “What about percent increase or percent decrease???” You’ll have to wait until Friday. (Promise. It’s not all that difficult either.)

This is a good thing to practice, so try out these problems. Remember: Identify the part, whole and percent before you use the proportion. (That’s not going to be as easy with these, because they’re not word problems.) Then cross multiply and get x by itself.

Questions about this process? Do you have any better ideas? (I’ll bet you do!) Share them in the comments section. Meanwhile, here are the answers to yesterday’s percent problems: 11/20, 41/50, 3/20, 0.04, 0.31, 1.4. How did you do?

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In Basic Math Review

Parts Is Parts: Get a handle on percents

Ever have one of those strings of bad ju-ju that just won’t quit? Welcome to the last two weeks of my life. From email woes to blog problems, it’s as if the electronic gods have cursed me. This is my way of explaining why there was no post yesterday. I’ll make it up to you today — as I wait on hold for the good folks at Comcast to answer my call about my email account. Wish me luck!

It’s the third week of our review of basic math. Time for percents. These little guys are everywhere — from the mall to your tax return to your kid’s grades to the nutritional label on your Cheerios. You simply cannot go a day without coming across a percent in one form or another.

(Try it. Just for today, notice the percents. If you’re so inclined, jot them down and post what you noticed in the comments section.)

So what’s the big deal? What are percents so darned ubiquitous?

Percents represent a part of the whole. We love to know what part of our extra-cheese, deep-dish pizza is fat or what part of the population is in favor of gun control. This information helps us make decisions and form opinions. And because of the way that percents are found, they’re not so challenging, actually.

First the basics: if you break down the word percent, you will immediately understand what it means. Per means every and (in the U.S.) a cent is 1/100 of a dollar. So percent literally means for every 1/100. Get it? (It should be noted that the notion of a percent came long before the U.S. penny, but the one-cent coin has its roots in Roman currency, which launched percents. Cool, huh?)

With this information, you can easily convert a percent to a fraction — which is a pretty darned useful thing to know. 10% is the same thing as 10 for every 1/100 or 10/100. The only thing left to do is simplify.

See what I did there? To turn a percent into a decimal, just put the percent over 100 and simplify. Works like a charm every single time.

But what about turning a percent into a decimal? That’s even easier. There are a couple of ways to look at this, but I chose 10% for a good reason. It’s the same thing as 1/10 or if you say it out-loud: “one-tenth.” And what’s another way of writing one-tenth? Put a decimal on it.

Think about what you learned in elementary school about decimals. One place to the left of the decimal point is the “tens” place. One to the right is the “tenths” place. Two places to the right is the “hundredths” place. And so on. If percents mean out of 100 or for every 1/100, really what you’re doing is thinking of place value.

10% = 0.10 = 0.1

All of this boils down to a really simple process. To change a percent to a decimal, move the decimal point two places to the right. Here are some examples:

Incredibly basic stuff, right? But it is important. We can use this information to help find the percent of a number or find the value of the whole, given the percent (which is a little bit harder). That’s up tomorrow and Friday.

Until then, how about giving these really simple problems a go?

Any of the above problems give you trouble? (Yep, I snuck in a few toughies, but I know you can do it. Just think it through.) Here are the answers to last Friday’s fraction problems: 2/3, 3/7, -3/14 (Yowza! That was a tricky one!), 5/9, 13/24.

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In Holidays

To Estimate or Not to Estimate: That is the question

Whether you’re buying gifts for under the tree or just taking advantage of holiday sales, December is one of those times when you might need some mental math skills.  And while it can seem overwhelming to find out how much that 15%-off cashmere sweater will actually cost you, there are some easy ways to make quick work of these calculations and move on to the next item on your to-do list.  (We’ll look at those on Friday.)

But first you need to answer one big question: Is an estimate good enough?

What’s the total cost?

Let’s say you’re picking up a few things for your Aunt Millie. She has given you a $20 bill and a list.  You absolutely cannot exceed $20, and Aunt Millie is adamant that you get as much as you can for that amount.  In this case, you may want to calculate everything down to the penny.

Or what if you’re purchasing holiday gifts for a family in need.  You’ve set your budget — and you’re not going over it!  Once you have everything in your cart, it could be reassuring to spend a moment or two finding the exact cost of your purchases.

(Here’s a cool hint, though.  If you’re shopping online, these calculations are done for you.  Just put what you want in your online shopping cart, and the totals will be appear — including shipping!)

Can you afford it?

But I would guess that most of us merely need to know if we can afford a purchase — or if what we’re interested in buying is too expensive.  And that’s where estimation comes in handy.

Chandra’s family is HUGE.  And after years of buying a Christmas gift for each of her nine siblings and their spouses and partners, she initiated the good old Secret Santa exchange.  What a relief!

The process is simple. Over pumpkin pie after Thanksgiving dinner, Chandra’s mother brings out her best Sunday hat, which contains slips of paper — one for each of the 18 kids and their partners.  Each person selects a name and buys a present for that person.  The catch? No one can spend more than $50.

This year, Chandra is over the moon.  She drew her sister-in-law’s name, and she knows exactly what to get her — a handmade purse from the local craft fair.

A week later, struggling through the crowd of candle-buying, carol-humming shoppers, Chandra finds exactly what she’s looking for: a cute little bag made of repurposed, 1940s dish towels.  What a find!

She snatches up the bag, and pays $40 for it.  But she’s got $10 left over.  Should she find something to put inside?

Chandra starts looking for a little something more: there’s a handmade key fob for $2.50 or a little zipper pouch for $10. She starts feeling like Goldilocks — the pouch is too much and the key fob is not enough.  She leaves knowing she can make up the difference while shopping elsewhere.

And she hits jackpot later that week.  While picking up a few things at her local, independent bookstore, she spies a sweet little journal at the checkout line that would just fit into the purse.  On sale for $6.50, she figures she has enough to pick up a rollerball pen to go with it.

Just right.  (And notice — very little math!)

Is estimation mandatory?

So let’s say you are really into knowing your costs down to the penny.  What if just having a general idea of what something costs is way too unnerving for you?

Pull out that calculator, sister or brother.  There’s nothing wrong with finding the exact answer, if that’s what you need or want to do.  Just do the rest of us shoppers a couple of favors — move to the side of the aisle while you do your computin’ and while you’re at it, don’t look down your nose at other’s estimations.

Are you an estimator or an exacting kind of person? If you estimate, how? If you like an exact answer, what tools do you use?  Share your stories in the comments section.

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