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Dividing by a Decimal
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Dividing by a decimal can look difficult, but actually there's only one simple step we need to do before we can divide by a decimal number. The following video will explain and show a couple of examples:

Dividing by a decimal

### Ways to show division

The division symbol (÷) is not the only way to show division. As you learned earlier in the mini-lesson about how to use Excel as a calculator, division can be symbolized with a slash (/) too. You may see this on some calculators as well. We will often use the / symbol within this course to represent division.

Remember, to divide by a decimal, multiply both numbers by the power of 10 that will make the divisor (the number we’re dividing by—for example, in $$5 \div 2.5$$, the divisor is $$2.5$$) an integer. Then the division process is exactly the same as when all the numbers were integers.

### Examples:

$$5 \div 2.5 =\text{?}$$

1. We multiply both numbers by $$10$$ to make it $$50 \div 25$$

$$88.2365 \div 3.69$$

1. We multiply both numbers by $$100$$, moving the decimal of the divisor over two spaces so it is an integer.
2. This makes the question $$8823.65 \div 369$$.
3. We just need the divisor to no longer have a decimal. The dividend can still have a decimal.

### Practice Problems

Evaluate the following expression:
1. $$54 \div 1.2 = \text{?}$$ (
Solution
Solution:
45
)
2. $$69 \div 2.4 = \text{?}$$ (
Solution
Solution: $$28.75$$
Details:
Move the decimal to the right one space in the divisor and dividend. (This is the same as multiplying both by 10.) The divisor is now a whole number 24 and the dividend is 690. Divide 69 by 24. The answer is about 2, since $$2 \times 24$$ is 48. Place the 48 below the 69 and subtract. The difference between 69 and 48 is 21. The arrow shows the next step, to bring down the zero. To continue dividing, divide 210 by 24.  $$8 \times 24$$ is 192. Place 192 below 210 and subtract. The difference between 210 and 192 is 18. Some regrouping was required to do the subtraction. We still have a remainder but there are not any more digits to bring down in the dividend. We place a decimal and place a 0 in the tenths place of the dividend so we have a 0 to bring down and continue our division process. 180 divided by 24 is about 7. Place the 7 in the top next to the 8, and leave a space for the decimal point which will be inserted at the end. $$24 × 7 = 168$$

Place 168 below 180 and subtract. The difference between 180 and 168 is 12. To continue, place another zero in the hundredths place of the dividend, The number will now show as 690.00. Bring down the new zero. Now divide 120 by 24

24 goes into 120 exactly 5 times.

Place 120 below the 120 and subtract. The difference is zero or no remainder so we can stop the division process. The answer isn’t complete, yet. Notice there is a space between 28 and 75. Place the decimal point in this space. The answer to this problem is $$28.75$$. There’s an arrow pointing from the decimal point in $$690.00$$ to $$28.75$$ to indicate where to place the decimal point. Note: It is important to keep all the numbers lined up neatly in their columns when doing division problems like this so you can keep track of things like the decimal point and bringing down the correct digits.
)
3. $$8.4 \div 6.5 = \text{?}$$ (Round your answer to the nearest hundredth.) (
Video Solution
Solution: $$1.29$$
Details:

(Video Source | Transcript)
)
4. $$24.7 \div 3.9 =\text{?}$$ (Round your answer to the nearest hundredth.) (
Video Solution
Solution: $$6.33$$
Details:

(Video Source | Transcript)
)
5. $$62.5 \div 1.75 =\text{?}$$ (Round your answer to the nearest hundredth.) (
Solution
Solution:
$$35.71$$
)
6. $$25 \div 4.35 = \text{?}$$ (Round your answer to the nearest hundredth.) (
Solution
Solution: $$5.75$$
Details:
Since the divisor in the problem has digits out to the hundredths place value, we need to multiply both the divisor and dividend by 100. This will make the divisor an integer but will not change the final answer of our division problem. We now have 2500 divided by 435. 435 does not go into 2 or 25 or even 250.

We need to go all the way out the 2500 before we can start making sets fo 435.

When dealing with larger numbers, sometimes we have to guess and then change our guess in order to find the correct numbers for our solution.

We are looking for a number, when multiplied to 435 that is close to 2500 but not more than it.

Guess #1: 7

$$435 \times 7 = 3045$$

Since 3045 is greater than 2500, 7 is too big. We need to guess a smaller number.

Guess #2: 6

$$435 \times 6 = 2610$$

This is still more than 2500, but it is very close to it. We probably just need to go down 1 more number.

Guess #3: 5

$$435 \times 5 = 2175$$

This is less than 2500 but still close to it.

Place the 5 in the answer location above the 0 in the ones place of 2500. Multiply $$5 \times 435 = 2175$$

Place 2175 below 2500 and subtract. $$2500 - 2175 = 325$$

Some regrouping is needed to do this subtraction. This means we have 325 left over, or remaining. This isn’t enough to make a group of 435 so we place a decimal point and another zero in the dividend and bring it down. We now can find the number of times 435 goes into 3250. Again, we use the guessing method.

Guess #1: 7

We guessed this before and know that $$435 \times 7 = 3045$$. This is close to 3250 but still less than it.

Place a 7 in the answer location next to the 5 and above the new 0. Multiply $$7 \times 435 = 3045$$ and place the 3045 below the 3250 and subtract. Do any regrouping needed. We now have 205 remaining. We need another 0 to continue our division process. Place another 0 in the hundredths place of the dividend and bring it down next to the 205 making it 2050. This time we guess that 435 goes into 2050 4 times.

Place 4 above the new 0 in the dividend and next to the 7.

Multiply $$4 \times 435 = 1740$$. Place 1740 below the 2050 and subtract. We still have a remainder so the process continues.

Bring another 0 down. Guess a close multiple of 435. Multiply that guess by 435. Subtract. Again, we see we still have a remainder left over. If we need to continue the division problem, then we continue, but at some point is usually sufficient to stop and round our final answer.

In this problem, we place our decimal point between the 5 and 7.

This means our solution is close to $$5.747$$…. The places after the decimal point continue beyond what we have solved. We have found enough digits in the solution to round to the nearest hundredth.

$$5.747$$… rounded to the nearest hundredth is $$5.75$$.

Our final solution is approximately: $$5.75$$
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