Estimation Problems using Fractions

When working with fractions, estimating can save time and effort while getting you very close to the correct answer. In this lesson, we'll learn how to use estimation to quickly solve problems involving fractions.

When Close Counts

Did you ever hear the saying 'Close only counts in horseshoes and hand grenades'? I immediately imagine people playing horseshoes with hand grenades, which is kind of dangerous.

But the point of that saying is not entirely true. There are a few other situations in which close counts just fine, like when I iron a shirt - it's close to wrinkle-free, and that's pretty good, or at least I think so. There's also baking cookies - close to done is when they're still gooey in the middle, which is actually better than being done in my opinion - or folding laundry, and especially a fitted sheet. I've never gotten better than close and my life seems to go on just fine.

Estimating Fractions

And then there are fractions. Fractions can be onerous. What if you have 5/322 and you need to add it to 420/863? You'd need to find the common denominator. (For the record, it's 277,886.)

But did you ever have 5/322nds of something? Let's say you're sharing pizza with some math geek friends and 5/322nds is your share. Do you know what you have? Maybe a small piece of one piece of pepperoni - in other words, pretty much zero.

And what about 420/863rds? If a glass can hold 863 drops of water and you put in 420, what do you have? A lot of free time, apparently. But you also have a glass that's about half full - or half empty. That's up to you. My point is that 420/863 is pretty close to 1/2.

What we just did is estimate the value of fractions in order to make them much simpler to work with. When you are estimating fractions, round the fraction to the nearest whole number, 1/2 or 1/4.

Practice Estimating

Here's what this looks like: You know 50/100 is 1/2. Think of 50/100 as $50 you're willing to spend out of the $100 you have. What if there's a dress you want to buy and it's $48? That's pretty much $50. With tax, it might even be a little more. And if it's $54? You might also think of that as being right around $50. You'd still be spending close to what you intended.

Let's consider another fraction: 60/80. That's 3/4. Well, 57/80 is also pretty much 3/4. And that saves you having to find a denominator like 277,886. Then you can get through your work way faster.

Practice Problems

Let's try a few practice problems. Let's say we want to add 84/90 to 32/29. 84/90 is very close to 90/90, or 1. And 32/29? That's also close to 1. So if we estimate the solution, it's going to be about 2. Note that if we didn't estimate, we'd need a denominator of 2610. We'd add 2436/2610 to 2880/2610 to get 5316/2610 to get 2 96/2610, or 2.04, which is super close to just 2. It's ever so slightly more than 2, so it's like a cookie that's just slightly burned. Some people love those burned-on-the-bottom cookies.

Here's one: 7 97/99 + 2 6/10. 97/99 is very close to 1, so let's make that first number 8. 6/10 is close to 1/2, so let's make that second number 2 1/2. 8 + 2 1/2 is 10 1/2.

What about 14/20 - 7/15? Well, 15/20 is 3/4, so we can estimate that 14/20 is nearly 3/4 as well. And 7/15? 7/14 is 1/2, so 7/15 is close to 1/2. Therefore, 14/20 - 7/15 is close to 3/4 - 1/2, or 1/4. I think this one's like a steak. If you wanted sort of medium rare, then that's what I can deliver - sort of medium rare. Sometimes it's more rare, sometimes it's more medium. But we're in the right ballpark.

Let's try a multiplication one. What is 11/23 * 3 5/9? This one's like parallel parking. You don't need to be right up next to the curb, but you don't want to be at a weird angle with one end of your car out in traffic. Close enough to the curb would be saying 11/23 is very close to 1/2. Sticking way out would be calling it 3/4. If you think in quarters, 11/23 is much closer to 1/2 than 3/4. And 5/9 is also close to 1/2. So 1/2 * 3 1/2 is 1 3/4.

How about 8/13 * 7/25? 8/13 is close to 1/2, but it's closer to 3/4. Why? 8/12 is 3/4. So 8/13 is just a little bigger than that. 7/25 is close to 1/4. What's 7 * 4? 28. So 7/28 is 1/4. 7/30 is just a little smaller. 3/4 * 1/4 is 3/16.

What about division? What is 72/150 divided by 37/75? Without estimating, those fractions are like fitted sheets. Seriously, though, I can't fold a fitted sheet. But I can estimate. And both of those fractions are close to 1/2. 1/2 divided by 1/2 is 1/2 * 2/1, or 1. With multiplication and division, the further you go from the exact numbers, the less accurate you get. Small differences are, well, multiplied. But this is still fairly accurate. Maybe this division problem is like hand-grenade horseshoes. You're throwing it in the general vicinity and, after the smoke clears, you probably earned yourself some points.

Lesson Summary

In summary, we learned that close counts in way more than horseshoes and hand grenades. Most importantly, estimating fractions is a great way to save time and effort.

The simplest way to estimate fractions is to use the closest whole number, 1/2 or 1/4.

For example, 18/19 is pretty much 1, 7/10 is close to 3/4 and 9/16 is very nearly 1/2.