Estimate Division with Two-Digit Divisors: Examples

How to Choose Compatible Numbers

Compatible numbers are a nearby pair you can divide mentally. Round the divisor to a convenient ten, then choose a nearby dividend that is a multiple of that divisor. Keep the original problem in view: changing the numbers gives an estimate, not an exact answer.

For a positive dividend, rounding it down while rounding a positive divisor up makes the quotient smaller. If both numbers move in the same direction, a multiplication check helps you judge the result. Avoid choosing a pair simply because it divides easily if it is far from the original numbers.

Worked Examples with Two-Digit Divisors

Example 1:

Divide \(525\) by \(46\).

Estimation Process:

1. Round \(525\) to \(500\).

2. Round \(46\) to \(50\).

3. Estimate the quotient: \(500 \div 50 = 10\).

Answer:

The estimated quotient is \(10\).

Check the estimate: \(46 \times 10 = 460\), which is below \(525\). The estimate of \(10\) is low; \(46 \times 11 = 506\) and \(46 \times 12 = 552\) show that the exact quotient lies between \(11\) and \(12\). A rough estimate need not be the nearest whole number.

Example 2:

Divide \(938\) by \(84\).

Estimation Process:

1. Round \(938\) to \(900\).

2. Round \(84\) to \(80\).

3. Using rounding gives \(900 \div 80 = 11.25\), or about \(11\). For a mental compatible-number estimate, choose \(880 \div 80 = 11\) instead.

Answer:

The estimated quotient is \(11\). Check: \(84 \times 11 = 924\) and \(84 \times 12 = 1008\), so the exact quotient is between \(11\) and \(12\).

The first example gives a rough lower estimate; the second gives an estimate close to the exact quotient. Decide how precise the task needs you to be, then check your choice by multiplication.

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Practice: Estimate Each Quotient

Choose nearby compatible numbers, write your estimate, and multiply to check. The sample answers below show one reasonable method; a different method can give a different estimate.

1. Estimate the quotient of \(672\) divided by \(56\).

2. What is the estimated quotient when \(1145\) is divided by \(92\)?

3. Divide \(789\) by \(72\) and estimate the quotient.

4. What is the estimated result of \(430\) divided by \(38\)?

5. Estimate the quotient of \(1256\) divided by \(45\).

Sample Estimates and Multiplication Checks

1. \(672 \div 56\): use \(720 \div 60 = 12\). Checking \(56 \times 12 = 672\) shows that this estimate happens to equal the exact quotient.

2. \(1145 \div 92\): use \(1080 \div 90 = 12\). Since \(92 \times 12 = 1104\) and \(92 \times 13 = 1196\), the exact quotient is between \(12\) and \(13\).

3. \(789 \div 72\): use \(770 \div 70 = 11\). Checking \(72 \times 11 = 792\) shows that \(11\) is a little high.

4. \(430 \div 38\): use \(440 \div 40 = 11\). Checking \(38 \times 11 = 418\) and \(38 \times 12 = 456\) places the exact quotient between \(11\) and \(12\).

5. \(1256 \div 45\): keep the friendly divisor \(45\) and use the nearby multiple \(1260\). Because \(45 \times 28 = 1260\), the estimate is \(28\), a little above the exact quotient.

Use an estimate to check the size of an answer. If a task requires an exact quotient or remainder, return to the original numbers and complete the division.

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