10.4: Estimation by Rounding

When beginning a computation, it is valuable to have an idea of what value to expect for the result. When a computation is completed, it is valuable to know if the result is reasonable.

In the rounding process, it is important to note two facts:

1. The rounding that is done in estimation does not always follow the rules of rounding discussed previously. Since estima­tion is concerned with the expected value of a computation, rounding is done using convenience as the guide rather than using hard-and-fast rounding rules. For example, if we wish to estimate the result of the division \(80 \div 26\) conveniently divided by 20 than by 30.
2. Since rounding may occur out of convenience, and different people have differ­ent ideas of what may be convenient, results of an estimation done by rounding may vary. For a particular computation, different people may get different estimated results. Results may vary.

Definition: Estimation

Estimation is the process of determining an expected value of a computation. Common words used in estimation are about, near, and between.

Estimation by Rounding

The rounding technique estimates the result of a computation by rounding the numbers involved in the computation to one or two nonzero digits.

Example \(\PageIndex{1}\)

Estimate the sum: \(2,357 + 6,106\).

Solution

Notice that 2,357 is near \(\underbrace{2,400}_{\text{two nonzero digits}}\) and that 6,106 is near \(\underbrace{6,100}_{\text{two nonzero digits}}\)

The sum can be estimated by \(2,400 + 6,100 = 8,500\). (It is quick and easy to add 24 and 61.)

Thus, \(2,357 + 6,106\) is about 8,400. In fact, \(2,357 + 6,106 = 8,463\).

Notice that the rounding happened before the operation was performed. Typically, a major reason we estimate is to make the computations easier. If you were to add \(2,357 + 6,106\) and then round the sum, you actually did more work than if you had just added \(2,357 + 6,106\) and written the answer. This means that sometimes the estimation is less accurate that we would get if we rounded after the computation. This is okay because we are only looking for estimates. This is one subtle difference between rounding and estimation. Rounding should reflect the reality of the value's size. Estimation might result in something quite different (though reasonable in the sense that it arose from reasonable methods).

Example \(\PageIndex{1}\)

Estimate each sum by rounding the addends.

• \(4,216 + 3,942\)
• \(812 + 514\)

You may use the \(\approx\) symbol to indicate that rounding has taken place.

Solution

• \(4,216 + 3,942 \approx 4,200 + 3,900\). So, \(4,216 + 3,942\) is about equal to 8,100.
• \(812 + 514 \approx 800 + 500\). So, \(812 + 514\) is about equal to 1,300.

For each case, find what the actual sum is and compare it to the estimate. How accurate are the estimates?

Try the following on your own. Compare the estimate you get with the actual sum.

Exercise \(\PageIndex{1}\)

Estimate the sum: \(43,892 + 92,106\)

Let's now estimate some differences.

Example \(\PageIndex{1}\)

Estimate the difference: \(5,203 - 3,015\).

Solution

Notice that 5,203 is near \(\underbrace{5,200}_{\text{two nonzero digits}}\) and that 3,015 is near \(\underbrace{3,000}_{\text{one nonzero digit}}\)

The difference can be estimated by \(5,200 - 3,000 = 2,200\).

Thus, \(5,203 - 3,015\) is about 2,200. In fact, \(5,203 - 3,015 = 2,188\).

We could make a less accurate estimation by observing that 5,203 is near 5,000. The number 5,000 has only one nonzero digit rather than two (as does 5,200). This fact makes the estimation quicker (but a little less accurate). We then estimate the difference by \(5,000 - 3,000 =2,000\). and conclude that \(5,203 - 3,015\) is about 2,000. This is why we say "answers may vary."

After reading the following example, redo it by with rounding to the tens place.

Example \(\PageIndex{1}\)

Estimate the difference: \(628 - 413\).

Solution

628 − 413 : 600 − 400. About 200. In fact, 215.

 Try the following on your own. If more than one method of rounding comes to mind, try them all. Compare your results to the actual differences.

Exercise \(\PageIndex{1}\)

Estimate the differences.

• \(7,842 - 5,209\)
• \(73,812 - 28,492\)

Let's try deal with multiplication now.

Example \(\PageIndex{1}\)

Estimate the product: \(73 \cdot 46\).

Solution

Notice that 73 rounded to the nearest ten is 70 and 46 rounded to the nearest ten is 50.

The product can be estimated by \(70 \cdot 50 = 3,500\).

Thus, \(73 \cdot 46\) is about 3,500. In fact, \(73 \cdot 46 = 3,358\).

Remember that we are using rounding to make these estimates, but that is not the only way to make estimates. If you thought of another method, that would be fine. For example, you might note that 73 is near 75 and 46 is near 40, despite 40 not being the nearest ten. Here's how one person used that method: 73 is near 75 and 46 is near 40, which is four tens. \(75\cdot 10\cdot 4 = 750\cdot 4 = 750 \cdot 2 \cdot 2 = 1500 \cdot 2 = 3000\). This requires a lot more comfort in "playing with numbers". So, if you're not feeling ready for this kind of mental math, that is okay. It takes time to develop number sense. Let's return to the rounding method of estimation.

Example \(\PageIndex{1}\)

Estimate the product: \(87 \cdot 4,316\).

Solution

We will again round each factor so that we only have one nonzero digit. Notice that 87 rounded to the nearest ten is 90 and 4,316 rounded to the nearest thousand is 4,000.

The product can be estimated by \(90 \cdot 4,000 = 360,000\).

Thus, \(87 \cdot 4,316\) is about 360,000. In fact, \(87 \cdot 4,316 = 375,492\).

Now try the following on your own.

Exercise \(\PageIndex{1}\)

Estimate the products.

1. \(31 \cdot 87\)
2. \(18 \cdot 421\)
3. \(160 \cdot 943\)

Let's try division.

Example \(\PageIndex{1}\)

Estimate the quotient: \(153 \div 17\).

Solution

Notice that 153 is close to \(\underbrace{150}_{\text{two nonzero digits}}\) and that 17 is close to \(\underbrace{15}_{\text{two nonzero digits}}\).

The quotient can be estimated by \(150 \div 15 = 10\).

Thus, \(153 \div 17\) is about 10. In fact, \(153 \div 17 = 9\).

The quotient in the previous example was made easy by the fact that 150 is a multiple of 10. Let's keep that in mind for the following example.

Example \(\PageIndex{1}\)

Estimate the quotient: \(742,000 \div 2,400\).

Solution

Let's look at two possibilities.

1. Notice that 742,000 is close to \(\underbrace{700,000}_{\text{one nonzero digit}}\) and that 2,400 is close to \(\underbrace{2,000}_{\text{one nonzero digit}}\).
   The quotient can be estimated by \(700,000 \div 2,000 = 350\). Thus, \(742,000 \div 2,400\) is about 350.
2. Notice that 742,000 is close to \(\underbrace{740,000}_{\text{two nonzero digits}}\) and that 2,400 is close to \(\underbrace{2,000}_{\text{one nonzero digit}}\).
   The quotient can be estimated by \(740,000 \div 2,000 = 370\). Thus, \(742,000 \div 2,400\) is about 370.

In fact, \(742,000 \div 2,400 = 309.1\overline{6}\).

Both were reasonable since 74 and 70 are multiples of 2. What if we changed the values just slightly, however?

Example \(\PageIndex{1}\)

Estimate the quotient: \(742,000 \div 2,900\).

Solution

Let's look at two possibilities.

1. Notice that 722,000 is close to \(\underbrace{700,000}_{\text{one nonzero digit}}\) and that 2,900 is close to \(\underbrace{3,000}_{\text{one nonzero digit}}\).
   The quotient can be estimated by \(700,000 \div 3,000\). This is not an easy calculation! So, let's change our tactic.
2. Notice that 722,000 is close to \(\underbrace{720,000}_{\text{two nonzero digits}}\) and that 2,900 is close to \(\underbrace{3,000}_{\text{one nonzero digit}}\).
   The quotient can be estimated by \(720,000 \div 3,000 = 240\). Thus, \(722,000 \div 2,900\) is about 240.

In addition to this showing how our choice of round-off digit is important, we can see how useful estimation is since actually \(722,000 \div 2,900 = 248.\overline{9655172413793103448275862068}\).

Try the following on your own. If your first try does not produce an easy calculation, try again.

Exercise \(\PageIndex{1}\)

Estimate the quotients.

1. \(221 \div 18\)
2. \(4,079 \div 381\)
3. \(609,000 \div 16,000\)