Mental Math: Estimation

Mike McGarry
Lesson by Mike McGarry
Magoosh Expert
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One helpful math technique on the ACT is estimation. Yes, you get a calculator, but doing the exact calculation is not necessary on every problem. Remember while the ACT asks you to perform calculations, it cares more about your ability to simplify problems with logic. So here's a question.

Pause the video, read this question, and then we'll talk about this. All right, so let's talk about this. For that problem, we could use the calculator to do an exact calculation, but that's not necessarily the best way. An exaction calculation would involve fractions or decimals because 21 is not divisible by 5.

But, we realize that 21 is very close to 20, which is divisible by 5. That will be a much quicker, neater calculation. So instead of a temperature 21, we're gonna say it's a temperature of 20. Well then, 20 / 5 becomes 4, that's very easy. 9 x 4 is 36. 36 + 32 is 68.

So that means that a Celsius temperature of 20 degrees corresponds to a Fahrenheit temperature of 68. So if it goes up 1 degree Celsius, then it would go up nine-fifths of a degree in Fahrenheit, so it would be something slightly bigger than 68. It would be something very close to 70, and that's the answer.

So notice that we could hone in on an answer without doing an exact calculation. Just by very simple approximation we could get that answer. With that problem, we are able to get the exact answer more easily simply by rounding to a close value. Sometimes, we can isolate a single answer. But even if we can't, estimating is an excellent way to eliminate answers and guess, especially if you don't know how to to do the problem or if you are running out of time.

So it's a very important strategy to keep up your sleeve. Here's another question. Pause the video and then we'll talk about this. Okay, when a company places a wholesale order on coffee cups with the company logo on it, it pays $692 for 80 cups. Well hm, that's kind of an ugly number, $692.

When Phil wants to buy just 1 cup, it costs him 12.50. How much is he paying above the wholesale price? So obviously there is a difference. Whatever $692 divided by 80 is, is gonna be less than $12.50. The company is making some kind of profit and we want to know about what is that profit.

So first of all, there's no way if he's only paying $12.50, that he's paying $14.25 above the wholesale price. The wholesale price, whatever it is, whatever $692 divided by 80 is, that's a positive number certainly. And so, there's no way that he's paying $14.25 over the wholesale price if he's only paying $12.50.

So, E is just an unreasonable answer. So what we're gonna do is we're gonna take that ugly number 692, and round it up a bit. We're gonna make a little more expensive, okay? In fact, the company paid a little bit less, so it's gonna make a little more of a profit.

But let's just figure out approximately what kind of profit would it make If it paid $720. And the reason I picked 720, it's very easy to divide that by 80. So 720 divided by 80 would be $9 per cup. Well, $9 per cup, if that's what the company is paying, in fact they're paying a little less than that.

But if they're paying $9 a cup, then that means they would be making a profit of around $3.50. Well in fact, they're paying a little bit less than $720, so it's a few pennies less than $9 a cup. So it's gonna be a few pennies more than $3.50. Well right away, that's enough to isolate what the answer's gonna be.

We know the answer will be B, $3.85. Here's another question. Pause the video, and then we'll talk about this. Okay. So to attend this dance, Sarah must buy shoes for $47, earrings for $16, and a ticket to the dance for $25.

There'll be an 8% tax on the shoes and earrings. What is the total amount Sarah must pay? Well first of all, let's just approximate, forget about the tax for a moment, let's just add up those three prices. We add up those three prices with no tax, that's $88. So with tax, it's gonna be a little more than that.

So, right away we know certainly she has to pay more than $88. So, the first two answers are just out. So even if we were running out of time where we didn't know how to do any other calculation beyond this, well right there, we can eliminate two answer choices. It strongly increases our odds of guessing correctly. But let's see, this is not too hard of a calculation.

And so we'll just look at this. So the part that we're paying tax on, that's the shoes plus the earring. That's the taxable part, that's $63. Well let's think about this. Figuring out 8% of that would be a bit of a pain in the butt. We don't necessarily wanna do that, but here's an easy way to think about it.

It's easy to figure out 10% of that. 10% percent of $63 is $6.30. Well, what's half of that? Divide that by 2. 5%, would have to be $3.15. So, we know that an 8% tax is gonna be somewhere between $3.15 and $6.30.

And so, if we add $3 to $88, that already gets us up to $91, so we have to be higher than $91. And so the only possible answer at this point is answer choice E. Sometimes, by the clever use of estimation, we can simplify a calculation and solve it without a calculator. Often, estimation allows us to eliminate one or two answer choices, and this can be very useful when we are guessing.

Estimation is also a very important skill to check with the reasonableness of your answers. Is it reasonable that the person would pay that much, that the temperature would be that high? Think in terms of real world references, is this something that makes sense in the real world?

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