Fluently Add, Subtract, Multiply, and Divide Multi-Digit Decimals: CCSS.Math.Content.6.NS.B.3 - 6th Grade Math

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Question

If Exam 1 is worth 25% of the total grade, Exam 2 is worth 25% and Exam 3 is worth 50%, what is Dave’s final grade?

Answer

Final Grade = Exam1 * 0.25 + Exam 2 * 0.25 + Exam3 * 0.50 =

960.25 + 700.25 + 85*0.5 =

24 + 17.5 + 42.5 = 84

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Question

A car has gas mileage of 46 miles per gallon. Assume that gas costs $3.90 per gallon. What would be the total cost of gas if the car traveled 1,150 miles?

Answer

First find out how many gallons of gas the car consumed while traveling 1150 miles. This can be found out like so:

1150 miles x gallons/46 miles = 25 gallons

Multiplying 25 by $3.90 yields $97.50.

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Question

Kevin is looking at the blueprint of the house he is building. The scale is 1 inch = 5 feet. On paper, the master bedroom is 2.5 inches by 3 inches. What is the actual size of the bedroom?

Answer

The given dimensions must be multiplied by 5, so 2.5 inches becomes 12.5 feet, and 3 inches becomes 15 feet.

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Question

A runner runs 10.2 miles east, then 2.3 miles west, then 1.4 miles east.

How many miles did the runner travel from where he started? (How far east did the runner go)?

Answer

When the runner is travelling east, it's in the positive direction. West is the negative direction. So we can compute it by doing the operation 10.2 + (–2.3) + 1.4 = 9.3

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Question

If , approximately how many inches are in one meter?

Answer

and $100\ cm=1\ m$

Using the factor-label method we get the following equation.

$\frac{1; meter}{1}\frac{100 ; cm}{1 ; m}\frac{1 ; in}{2.5; cm} = 40 ; in$

We treat the units as if they are numbers, and can cancel the units from the fractions.

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Question

Fill in the blank to make a true statement:

_______ $- \ 0.7 = 2.8$

Answer

To find out what number to subtract 0.7 from to get 2.8, just do the opposite - add 0.7 to 2.8.

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Question

What number is in the hundredths place of

Answer

. The hundredths place is occupied by 7.

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Question

The menu of a local coffeehouse reads as follows:

$\begin{matrix} \textrm{Espresso}; ; ; & $1.79 \ \textrm{Cafe Latte} & $2.19 \ \textrm{Cappucino} & $2.29 \ \textrm{Americano} & $2.39 \ \textrm{Turkish; ; } & $2.09 \ \textrm{Iced Tea}; ; ; & $1.59 \end{matrix}$

Sandy orders some drinks for herself and some friends. She orders three cappuccinos, two iced teas, two cafe lattes, and an espresso. The sales tax is five percent. How much change does she receive back for a twenty-dollar bill?

Answer

The three cappuccinos cost: $$2.29 \times 3 = $6.87$

The two iced teas cost $$1.59 \times 2 = $3.18$

The two cafe lattes cost $$2.19 \times 2 = $4.38$

The espresso costs .

Add these amounts to get the cost before tax:

The tax is five percent of or:

$$16.22 \times 0.05 \approx $ 0.81$

Add both values in order to obtain the total cost after tax.

As a result, the change from a twenty-dollar bill is as follows:

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Question

Dawna wants to buy coffee beans from a local supermarket. If the coffee beans costs a pound, how much would Dawna have to spend on coffee beans if she only wants to buy $\frac{1}{4}$ of a pound?

Answer

We have to multiply the cost per pound by the weight that Dawna wants:

$10 \times \frac{1}{4}= 2.50$

This is the same as saying: $$10\div 4 = $2.50$ .

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Question

Solve for :

Answer

$0.15 h \div 0.15 = 18\div 0.15$

$h = 18\div 0.15$

To divide, move the decimal point in both numbers right two places to make the divisor whole.

$h = 1,800\div 15 = 120$

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Question

You bought a dozen eggs marked at and received change from . What is the percent of sales tax?

Answer

Set the equation up as

Solve for , which equals $\frac{2.45}{2.29}$

or

Therefore the percent sales tax is:

$(1.0698-1) \times 100=6.98% \approx 7%$

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Question

Define an operation $\bigstar$ on the set of positive integers as follows:

$a \bigstar b = GCF (a,b)$

If $a \bigstar 20 = 10$ , then which of the following could not be equal to?

Answer

If $a \bigstar 20 = 10$ , then the greatest common factor of and 20 is 10. 80 and 20 both have 20 as a factor - 20 is its own factor, since $20 \div 20 = 1$ , and 20 is a factor of 80, since $80 \div 20 = 4$ . Therefore, 10 is not the greatest common factor of 20 and 80, and 80 is the correct choice.

Of the other three factors, it can be seen that the GCF of 20 and each individual number is 10 by looking at each prime factorization:

$20 = \underline{2} \cdot 2 \cdot \underline{5}$

$30 = \underline{2} \cdot 3 \cdot \underline{5}$

$50 = \underline{2} \cdot 5 \cdot \underline{5}$

$70 = \underline{2} \cdot \underline{5} \cdot 7$

In each pairing, the common prime factors are 2 and 5, so the GCF is $2 \cdot 5 = 10$ .

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Question

Above is the menu at the concession stand at a school carnival.

David wants to buy some chicken strips for himself and his companions. He wants six strips, his brother Mickey wants five strips, his girlfriend Vickie wants three, and Mickey's girlfriend Jenny wants five. Since the concession stand only sells the strips in orders of four, he has decided to just purchase as many orders as it takes to feed the entire group. How much will he spend on enough orders of strips to satisfy the group?

(No tax is charged, since this is a school event)

Answer

The group wants a total of strips; since

$19 \div 4 = 4 \textup{ R }3$ ,

Dave will need to purchase five orders of strips (he cannot purchase a partial order). This will cost him

$$3.99 \times 5 = $19.95$ .

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Question

The above is a menu at the concession stand at a drive-in movie.

Gary wants to order two hamburgers and two small orders of fries. He wants to order two sodas of the same size. If he has just a twenty-dollar bill on hand, what is the largest size soda of which he can order two?

Answer

The easiest way to think of this is to note that Gary seems to be ordering for two people, himself and a friend. He has $20, so half of this will be for himself and half for his friend - and half of $20 is $10.

Each hamburger costs $4.89, and each small order of fries costs $2.29. This leaves

to spend on a soda for each person. This will enable to him to buy both himself and his friend a large soda.

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Question

Solve:

${0.42{\overline{\smash{)}4.19348}}}$

Answer

The first thing that we want to do when dividing decimals is to turn the divisor into a whole number. We do this by moving the decimal place to the right:

${0.42\rightarrow 42}$

If we move the decimal over two places in the divisor, we must also move the decimal over two places in the dividend:

${4.19348\rightarrow 419.348}$

The new division problem should look as follows:

$\ \ \ \ .\42{\overline{\smash{)}419.348}}$

*Notice how we've already placed the decimal in our answer. When we divide decimals, we place the decimal directly above the decimal in the dividend, but only after we've completed the first two steps of moving the decimal point in the divisor and dividend.

Now we can divide like normal:

$\ \ \ \ .\42{\overline{\smash{)}419.348}}$

Think: how many times can 42 go into 419

42 can go into 419 nine times times so we write a 9 over the 9 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000} 9.~\phantom{~~0} } \[-3pt]42\ )419.348\ \end{array}$

Next, we multiply 9 and 42 and write that product underneath the 419 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}9.~\phantom{~~0} } \[-3pt] 42) \ 419.348\ \underline{-378\phantom{~000}} \41\phantom{~000} \end{array}$

Now we bring down the 3 from the dividend to make the 41 into a 413.

Think: how many times can 42 go into 413?

42 can go into 413 nine times so we write a 9 above the 3 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}9.9\phantom{00} } \[-3pt]42) \ 419.348\ \underline{-378\phantom{~000}} \ 413\phantom{~00} \end{array}$

Next, we multiply 9 and 42 and write that product underneath the 413 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}9.9 \phantom{00} } \[-3pt]42) \ 419.348 \ \underline{-378\phantom{~000}} \ 413 \phantom{~00} \ \underline{-378\phantom{~00}} \35\phantom{~00} \end{array}$

Now we bring down the 4 from the dividend to make the 35 into a 354.

Think: how many times can 42 go into 354?

42 can go into 354 eight times so we write a 8 above the 4 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}9.98 \phantom{0} } \[-3pt]42) \ 419.348 \ \underline{-378\phantom{~000}} \ 413\phantom{~00} \ \underline{-378\phantom{~00}} \ 354 \phantom{~0} \end{array}$

Next, we multiply 8 and 42 and write that product underneath the 354 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}9.98 \phantom{0} } \[-3pt]42) \ 419.348 \ \underline{-378\phantom{~000}} \ 413\phantom{~00} \ \underline{-378\phantom{~00}} \ 354 \phantom{~0} \ \underline{-336\phantom{~0}} \18 \phantom{~0} \end{array}$

Now we bring down the 8 from the dividend to make the 18 into a 188.

Think: how many times can 42 go into 188?

42 can go into 188 four times so we write a 4 above the 8 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}9.984 } \[-3pt]42) \ 419.348 \ \underline{-378\phantom{~000}} \ 413\phantom{~00} \ \underline{-378\phantom{~00}} \ 354 \phantom{~0} \ \underline{-336\phantom{~0}} \188 \phantom{~} \end{array}$

Next, we multiply 4 and 42 and write that product underneath the 188 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}9.984 } \[-3pt]42) \ 419.348 \ \underline{-378\phantom{~000}} \ 413\phantom{~00} \ \underline{-378\phantom{~00}} \ 354 \phantom{~0} \ \underline{-336\phantom{~0}} \188 \phantom{~} \ \underline{-168\phantom{~}} \ 20\phantom{~} \end{array}$

Notice our remainder is 20 so our answer is 9.984R20.

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Question

Solve:

${0.88{\overline{\smash{)}3.19851}}}$

Answer

The first thing that we want to do when dividing decimals is to turn the divisor into a whole number. We do this by moving the decimal place to the right:

${0.88\rightarrow 88}$

If we move the decimal over two places in the divisor, we must also move the decimal over two places in the dividend:

${3.19851\rightarrow 319.851}$

The new division problem should look as follows:

$\ \ \ \ .\88{\overline{\smash{)}319.851}}$

*Notice how we've already placed the decimal in our answer. When we divide decimals, we place the decimal directly above the decimal in the dividend, but only after we've completed the first two steps of moving the decimal point in the divisor and dividend.

Now we can divide like normal:

$\ \ \ \ .\88{\overline{\smash{)}319.851}}$

Think: how many times can 88 go into 319

88 can go into 319 three times times so we write a 3 over the 9 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000} 3.~\phantom{~~0} } \[-3pt]88\ )319.851\ \end{array}$

Next, we multiply 3 and 88 and write that product underneath the 319 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}3.~\phantom{~~0} } \[-3pt] 88) \ 319.851\ \underline{-264\phantom{~000}} \55\phantom{~000} \end{array}$

Now we bring down the 8 from the dividend to make the 55 into a 558.

Think: how many times can 88 go into 558?

88 can go into 558 six times so we write a 6 above the 8 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}3.6\phantom{00} } \[-3pt]88) \ 319.851\ \underline{-264\phantom{~000}} \ 558\phantom{~00} \end{array}$

Next, we multiply 6 and 88 and write that product underneath the 558 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}3.6 \phantom{00} } \[-3pt]88) \ 319.851 \ \underline{-264\phantom{~000}} \ 558 \phantom{~00} \ \underline{-528\phantom{~00}} \30\phantom{~00} \end{array}$

Now we bring down the 5 from the dividend to make the 30 into a 305.

Think: how many times can 88 go into 305?

88 can go into 305 three times so we write a 3 above the 5 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}3.63 \phantom{0} } \[-3pt]88) \ 319.851 \ \underline{-264\phantom{~000}} \ 558\phantom{~00} \ \underline{-528\phantom{~00}} \ 305 \phantom{~0} \end{array}$

Next, we multiply 3 and 88 and write that product underneath the 305 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}3.63 \phantom{0} } \[-3pt]88) \ 319.851 \ \underline{-264\phantom{~000}} \ 558\phantom{~00} \ \underline{-528\phantom{~00}} \ 305 \phantom{~0} \ \underline{-264\phantom{~0}} \41 \phantom{~0} \end{array}$

Now we bring down the 1 from the dividend to make the 41 into a 411.

Think: how many times can 88 go into 411?

88 can go into 411 four times so we write a 4 above the 1 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}3.634 } \[-3pt]88) \ 319.851 \ \underline{-264\phantom{~000}} \ 558\phantom{~00} \ \underline{-528\phantom{~00}} \ 305 \phantom{~0} \ \underline{-264\phantom{~0}} \411 \phantom{~} \end{array}$

Next, we multiply 4 and 88 and write that product underneath the 411 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}3.634 } \[-3pt]88) \ 319.851 \ \underline{-264\phantom{~000}} \ 558\phantom{~00} \ \underline{-528\phantom{~00}} \ 305 \phantom{~0} \ \underline{-264\phantom{~0}} \411 \phantom{~} \ \underline{-352\phantom{~}} \ 59\phantom{~} \end{array}$

Notice our remainder is 59 so our answer is 3.634R59.

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Question

Solve:

${0.64{\overline{\smash{)}3.00216}}}$

Answer

The first thing that we want to do when dividing decimals is to turn the divisor into a whole number. We do this by moving the decimal place to the right:

${0.64\rightarrow 64}$

If we move the decimal over two places in the divisor, we must also move the decimal over two places in the dividend:

${3.00216\rightarrow 300.216}$

The new division problem should look as follows:

$\ \ \ \ .\64{\overline{\smash{)}300.216}}$

*Notice how we've already placed the decimal in our answer. When we divide decimals, we place the decimal directly above the decimal in the dividend, but only after we've completed the first two steps of moving the decimal point in the divisor and dividend.

Now we can divide like normal:

$\ \ \ \ .\64{\overline{\smash{)}300.216}}$

Think: how many times can 64 go into 300

64 can go into 300 four times times so we write a 4 over the 0 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000} 4.~\phantom{~~0} } \[-3pt]64\ )300.216\ \end{array}$

Next, we multiply 4 and 64 and write that product underneath the 300 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}4.~\phantom{~~0} } \[-3pt] 64) \ 300.216\ \underline{-256\phantom{~000}} \44\phantom{~000} \end{array}$

Now we bring down the 2 from the dividend to make the 44 into a 442.

Think: how many times can 64 go into 442?

64 can go into 442 six times so we write a 6 above the 2 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}4.6\phantom{00} } \[-3pt]64) \ 300.216\ \underline{-256\phantom{~000}} \ 442\phantom{~00} \end{array}$

Next, we multiply 6 and 64 and write that product underneath the 442 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}4.6 \phantom{00} } \[-3pt]64) \ 300.216 \ \underline{-256\phantom{~000}} \ 442 \phantom{~00} \ \underline{-384\phantom{~00}} \58\phantom{~00} \end{array}$

Now we bring down the 1 from the dividend to make the 58 into a 581.

Think: how many times can 64 go into 581?

64 can go into 581 nine times so we write a 9 above the 1 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}4.69 \phantom{0} } \[-3pt]64) \ 300.216 \ \underline{-256\phantom{~000}} \ 442\phantom{~00} \ \underline{-384\phantom{~00}} \ 581 \phantom{~0} \end{array}$

Next, we multiply 9 and 64 and write that product underneath the 581 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}4.69 \phantom{0} } \[-3pt]64) \ 300.216 \ \underline{-256\phantom{~000}} \ 442\phantom{~00} \ \underline{-384\phantom{~00}} \ 581 \phantom{~0} \ \underline{-576\phantom{~0}} \5 \phantom{~0} \end{array}$

Now we bring down the 6 from the dividend to make the 5 into a 56.

Think: how many times can 64 go into 56?

64 can go into 56 zero so we write a 0 above the 6 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}4.690 } \[-3pt]64) \ 300.216 \ \underline{-256\phantom{~000}} \ 442\phantom{~00} \ \underline{-384\phantom{~00}} \ 581 \phantom{~0} \ \underline{-576\phantom{~0}} \56 \phantom{~} \end{array}$

Next, we multiply 0 and 64 and write that product underneath the 56 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}4.690 } \[-3pt]64) \ 300.216 \ \underline{-256\phantom{~000}} \ 442\phantom{~00} \ \underline{-384\phantom{~00}} \ 581 \phantom{~0} \ \underline{-576\phantom{~0}} \56 \phantom{~} \ \underline{-0\phantom{~}} \ 56\phantom{~} \end{array}$

Notice our remainder is 56 so our answer is 4.690R56.

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Question

Solve:

${0.82{\overline{\smash{)}1.99402}}}$

Answer

The first thing that we want to do when dividing decimals is to turn the divisor into a whole number. We do this by moving the decimal place to the right:

${0.82\rightarrow 82}$

If we move the decimal over two places in the divisor, we must also move the decimal over two places in the dividend:

${1.99402\rightarrow 199.402}$

The new division problem should look as follows:

$\ \ \ \ .\82{\overline{\smash{)}199.402}}$

*Notice how we've already placed the decimal in our answer. When we divide decimals, we place the decimal directly above the decimal in the dividend, but only after we've completed the first two steps of moving the decimal point in the divisor and dividend.

Now we can divide like normal:

$\ \ \ \ .\82{\overline{\smash{)}199.402}}$

Think: how many times can 82 go into 199

82 can go into 199 two times times so we write a 2 over the 9 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000} 2.~\phantom{~~0} } \[-3pt]82\ )199.402\ \end{array}$

Next, we multiply 2 and 82 and write that product underneath the 199 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}2.~\phantom{~~0} } \[-3pt] 82) \ 199.402\ \underline{-164\phantom{~000}} \35\phantom{~000} \end{array}$

Now we bring down the 4 from the dividend to make the 35 into a 354.

Think: how many times can 82 go into 354?

82 can go into 354 four times so we write a 4 above the 4 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}2.4\phantom{00} } \[-3pt]82) \ 199.402\ \underline{-164\phantom{~000}} \ 354\phantom{~00} \end{array}$

Next, we multiply 4 and 82 and write that product underneath the 354 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}2.4 \phantom{00} } \[-3pt]82) \ 199.402 \ \underline{-164\phantom{~000}} \ 354 \phantom{~00} \ \underline{-328\phantom{~00}} \26\phantom{~00} \end{array}$

Now we bring down the 0 from the dividend to make the 26 into a 260.

Think: how many times can 82 go into 260?

82 can go into 260 three times so we write a 3 above the 0 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}2.43 \phantom{0} } \[-3pt]82) \ 199.402 \ \underline{-164\phantom{~000}} \ 354\phantom{~00} \ \underline{-328\phantom{~00}} \ 260 \phantom{~0} \end{array}$

Next, we multiply 3 and 82 and write that product underneath the 260 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}2.43 \phantom{0} } \[-3pt]82) \ 199.402 \ \underline{-164\phantom{~000}} \ 354\phantom{~00} \ \underline{-328\phantom{~00}} \ 260 \phantom{~0} \ \underline{-246\phantom{~0}} \14 \phantom{~0} \end{array}$

Now we bring down the 2 from the dividend to make the 14 into a 142.

Think: how many times can 82 go into 142?

82 can go into 142 one time so we write a 1 above the 2 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}2.431 } \[-3pt]82) \ 199.402 \ \underline{-164\phantom{~000}} \ 354\phantom{~00} \ \underline{-328\phantom{~00}} \ 260 \phantom{~0} \ \underline{-246\phantom{~0}} \142 \phantom{~} \end{array}$

Next, we multiply 1 and 82 and write that product underneath the 142 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}2.431 } \[-3pt]82) \ 199.402 \ \underline{-164\phantom{~000}} \ 354\phantom{~00} \ \underline{-328\phantom{~00}} \ 260 \phantom{~0} \ \underline{-246\phantom{~0}} \142 \phantom{~} \ \underline{-82\phantom{~}} \ 60\phantom{~} \end{array}$

Notice our remainder is 60 so our answer is 2.431R60.

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Question

Solve:

${0.44{\overline{\smash{)}3.44256}}}$

Answer

The first thing that we want to do when dividing decimals is to turn the divisor into a whole number. We do this by moving the decimal place to the right:

${0.44\rightarrow 44}$

If we move the decimal over two places in the divisor, we must also move the decimal over two places in the dividend:

${3.44256\rightarrow 344.256}$

The new division problem should look as follows:

$\ \ \ \ .\44{\overline{\smash{)}344.256}}$

*Notice how we've already placed the decimal in our answer. When we divide decimals, we place the decimal directly above the decimal in the dividend, but only after we've completed the first two steps of moving the decimal point in the divisor and dividend.

Now we can divide like normal:

$\ \ \ \ .\44{\overline{\smash{)}344.256}}$

Think: how many times can 44 go into 344

44 can go into 344 seven times times so we write a 7 over the 4 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000} 7.~\phantom{~~0} } \[-3pt]44\ )344.256\ \end{array}$

Next, we multiply 7 and 44 and write that product underneath the 344 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}7.~\phantom{~~0} } \[-3pt] 44) \ 344.256\ \underline{-308\phantom{~000}} \36\phantom{~000} \end{array}$

Now we bring down the 2 from the dividend to make the 36 into a 362.

Think: how many times can 44 go into 362?

44 can go into 362 eight times so we write a 8 above the 2 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}7.8\phantom{00} } \[-3pt]44) \ 344.256\ \underline{-308\phantom{~000}} \ 362\phantom{~00} \end{array}$

Next, we multiply 8 and 44 and write that product underneath the 362 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}7.8 \phantom{00} } \[-3pt]44) \ 344.256 \ \underline{-308\phantom{~000}} \ 362 \phantom{~00} \ \underline{-352\phantom{~00}} \10\phantom{~00} \end{array}$

Now we bring down the 5 from the dividend to make the 10 into a 105.

Think: how many times can 44 go into 105?

44 can go into 105 two times so we write a 2 above the 5 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}7.82 \phantom{0} } \[-3pt]44) \ 344.256 \ \underline{-308\phantom{~000}} \ 362\phantom{~00} \ \underline{-352\phantom{~00}} \ 105 \phantom{~0} \end{array}$

Next, we multiply 2 and 44 and write that product underneath the 105 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}7.82 \phantom{0} } \[-3pt]44) \ 344.256 \ \underline{-308\phantom{~000}} \ 362\phantom{~00} \ \underline{-352\phantom{~00}} \ 105 \phantom{~0} \ \underline{-88\phantom{~0}} \17 \phantom{~0} \end{array}$

Now we bring down the 6 from the dividend to make the 17 into a 176.

Think: how many times can 44 go into 176?

44 can go into 176 three times so we write a 3 above the 6 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}7.823 } \[-3pt]44) \ 344.256 \ \underline{-308\phantom{~000}} \ 362\phantom{~00} \ \underline{-352\phantom{~00}} \ 105 \phantom{~0} \ \underline{-88\phantom{~0}} \176 \phantom{~} \end{array}$

Next, we multiply 3 and 44 and write that product underneath the 176 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}7.823 } \[-3pt]44) \ 344.256 \ \underline{-308\phantom{~000}} \ 362\phantom{~00} \ \underline{-352\phantom{~00}} \ 105 \phantom{~0} \ \underline{-88\phantom{~0}} \176 \phantom{~} \ \underline{-132\phantom{~}} \ 44\phantom{~} \end{array}$

Notice our remainder is 44 so our answer is 7.823R44.

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Question

Solve:

${0.84{\overline{\smash{)}1.59708}}}$

Answer

The first thing that we want to do when dividing decimals is to turn the divisor into a whole number. We do this by moving the decimal place to the right:

${0.84\rightarrow 84}$

If we move the decimal over two places in the divisor, we must also move the decimal over two places in the dividend:

${1.59708\rightarrow 159.708}$

The new division problem should look as follows:

$\ \ \ \ .\84{\overline{\smash{)}159.708}}$

*Notice how we've already placed the decimal in our answer. When we divide decimals, we place the decimal directly above the decimal in the dividend, but only after we've completed the first two steps of moving the decimal point in the divisor and dividend.

Now we can divide like normal:

$\ \ \ \ .\84{\overline{\smash{)}159.708}}$

Think: how many times can 84 go into 159

84 can go into 159 one time times so we write a 1 over the 9 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000} 1.~\phantom{~~0} } \[-3pt]84\ )159.708\ \end{array}$

Next, we multiply 1 and 84 and write that product underneath the 159 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}1.~\phantom{~~0} } \[-3pt] 84) \ 159.708\ \underline{-84\phantom{~000}} \75\phantom{~000} \end{array}$

Now we bring down the 7 from the dividend to make the 75 into a 757.

Think: how many times can 84 go into 757?

84 can go into 757 nine times so we write a 9 above the 7 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}1.9\phantom{00} } \[-3pt]84) \ 159.708\ \underline{-84\phantom{~000}} \ 757\phantom{~00} \end{array}$

Next, we multiply 9 and 84 and write that product underneath the 757 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}1.9 \phantom{00} } \[-3pt]84) \ 159.708 \ \underline{-84\phantom{~000}} \ 757 \phantom{~00} \ \underline{-756\phantom{~00}} \1\phantom{~00} \end{array}$

Now we bring down the 0 from the dividend to make the 1 into a 10.

Think: how many times can 84 go into 10?

84 can go into 10 zero so we write a 0 above the 0 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}1.90 \phantom{0} } \[-3pt]84) \ 159.708 \ \underline{-84\phantom{~000}} \ 757\phantom{~00} \ \underline{-756\phantom{~00}} \ 10 \phantom{~0} \end{array}$

Next, we multiply 0 and 84 and write that product underneath the 10 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}1.90 \phantom{0} } \[-3pt]84) \ 159.708 \ \underline{-84\phantom{~000}} \ 757\phantom{~00} \ \underline{-756\phantom{~00}} \ 10 \phantom{~0} \ \underline{-0\phantom{~0}} \10 \phantom{~0} \end{array}$

Now we bring down the 8 from the dividend to make the 10 into a 108.

Think: how many times can 84 go into 108?

84 can go into 108 one time so we write a 1 above the 8 in the dividend:

$\begin{array}{rl} \underline{ \phantom{000}1.901 } \[-3pt]84) \ 159.708 \ \underline{-84\phantom{~000}} \ 757\phantom{~00} \ \underline{-756\phantom{~00}} \ 10 \phantom{~0} \ \underline{-0\phantom{~0}} \108 \phantom{~} \end{array}$

Next, we multiply 1 and 84 and write that product underneath the 108 and subtract:

$\begin{array}{rl} \underline{ \phantom{000}1.901 } \[-3pt]84) \ 159.708 \ \underline{-84\phantom{~000}} \ 757\phantom{~00} \ \underline{-756\phantom{~00}} \ 10 \phantom{~0} \ \underline{-0\phantom{~0}} \108 \phantom{~} \ \underline{-84\phantom{~}} \ 24\phantom{~} \end{array}$

Notice our remainder is 24 so our answer is 1.901R24.

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