# Estimation

We **estimate** when we do not need an exact answer but just one that is close enough. Some questions will ask us to find the "closest approximation" or "the best estimate." We approach these problems by identifying the term with the most significant impact, and setting up a simple calculation; we then improve our approximation if necessary.

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## Introduction

Estimation can help us identify potential errors and quickly double-check our answers. If by estimating we are able to narrow down the range of the answers in a problem, that will allow us to eliminate wrong choices and improve our chances at guessing. If we can eliminate all but one possibility, then that must be the correct answer!

In general, estimating involves taking key quantities in the problem and rounding them off so that they can be manipulated easier. For example, if you are trying to multiply together \(41\) and \(69\), you might round one up and one down to \(40\) and \(70\) respectively. Then you can very quickly come up with an estimation of \(2800\). As the actual number is \(2829\), this is a reasonable estimate. The same type of rounding can be applied to adding, subtracting, or dividing numbers as well.

In the next section, you will see a few examples of how and when rounding can be useful.

## Examples And Problems

If Mary has 41 quarters in her pocket, approximately how much money does she have?

(A) \(\ \ $0.40\)

(B) \(\ \ $2\)

(C) \(\ \ $4\)

(D) \(\ \ $10\)

(E) \(\ \ $40\)

Correct Answer: D

Solution:She has approximately \( 41 \times \frac{1}{4} = \frac{41}{4}=10 \frac{1}{4} \approx10 \) dollars in her pocket.

Incorrect Choices:

(A)

If you think Mary has 41 pennies in her pocket, you will get this wrong answer.

(B)

If you think Mary has 41 nickels in her pocket, you will get this wrong answer.

(C)

If you think Mary has 41 dimes in her pocket, you will get this wrong answer.

(E)

If you think Mary has 41 dollar coins in her pocket, you will get this wrong answer.

If one meal costs $5.88, which of the following is the best approximation for the cost of eight meals?

(A) \(\ \ $40.00\)

(B) \(\ \ $48.00\)

(C) \(\ \ $50.00\)

(D) \(\ \ $58.00\)

(E) \(\ \ $60.00\)

Correct Answer: B

Solution:We round the cost of the meal up to $6.00 and obtain

\[ 8 \times $5.88 \approx 8 \times $6.00 = $48.00. \]

Note that since we rounded the price up, this is an over-approximation, and the actual cost is below $48.00.

Incorrect Choices:

(A)

If you only take into account the dollar part ($5.00) of the price, you will get this wrong answer. However, 88 cents is pretty close to a dollar, and the actual price will be significantly higher than this approximation.

(C)

If you estimated $48.00, and then rounded your estimation to $50.00, you will get this wrong answer. $48.00 is a better approximation than $50.00 because it is closer to the actual value of the purchase.

(D)

If you rounded the number of meals from 8 to 10, and then rounded \(10 \cdot $5.88 = $58.80\) down to $58.00, you will get this wrong answer. Since the number of meals is so small, every meal will contribute a substantial amount to the final cost. So, it is better to round the price than the number of meals.

(E)

If you rounded the number of meals from 8 to 10, and then rounded \(10 \cdot $5.88 = $58.80\) to $60.00, you will get this wrong answer. Since the number of meals is so small, every meal will contribute a substantial amount to the final cost. So, it is better to round the price than the number of meals.If you got this problem wrong, you should review Algebraic Manipulations.

If a pen costs $1.89 and an eraser costs $0.99, which of the following is the best estimate for the cost of 10 pens and 5 erasers?

(A) \(\ \ $10.00\)

(B) \(\ \ $15.00\)

(C) \(\ \ $20.00\)

(D) \(\ \ $25.00\)

(E) \(\ \ $30.00\)

Correct Answer: D

Solution:We can round the price of a pen up from $1.89 to $2.00, and also round up the price of an eraser from $0.99 to $1.00. Then we get:

\[10 \times $1.89 + 5 \times $0.99 \approx 10 \times $2.00 + 5 \times $1.00 = $25.00.\]

Incorrect Choices:

(A)

If you round the price of a pen down to $1.00 and the price of an eraser down to $0.00, you will get this wrong answer.

(B)

If you round the price of a pen down to $1.00 and the price of an eraser up to $1.00, you will get this wrong answer.

(C)

If you round the price of a pen up to $2.00 and the price of an eraser down to $0.00, you will get this wrong answer.

(E)

If you round the estimation of $25.00 up to $30.00, you will get this wrong answer. There is no need to round up the initial approximation.