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All four operations on large numbers and decimals, done reliably.
Paper packet. Every task here also exists on screen, where it is checked automatically; answers written on paper are not assessed by Nydus. When you are back at a device, enter your answers there.
In this lesson you calculate quickly and accurately with multi-digit numbers and decimals in all four operations: multiplying by a two-digit number, long division by a two-digit number, and adding, subtracting, multiplying and dividing decimals. Fluency matters because every later topic leans on it: an equation is hard enough without the arithmetic inside it also being hard.
You know your multiplication facts up to $9 \times 9$, and you can multiply a number by 10 or 100 by writing zeros. In grade 5 you multiplied whole numbers in columns and divided with one-digit and two-digit divisors, and you added and subtracted decimals to hundredths. This lesson makes all of that fast and reliable, and adds the standard way to divide by a two-digit number and to divide by a decimal.
| Term | What it means |
|---|---|
| Dividend | The number being divided. In $3528 \div 42$ it is 3,528. |
| Divisor | The number you divide by. In $3528 \div 42$ it is 42. |
| Quotient | The answer to a division. $3528 \div 42 = 84$, so the quotient is 84. |
| Remainder | What is left over when the divisor does not fit exactly. |
| Partial product | One piece of a multiplication, such as $316 \times 4$ inside $316 \times 24$, before the pieces are added. |
| Decimal place | A digit to the right of the decimal point: tenths, hundredths, thousandths. |
| Estimate | A rough answer, found quickly with rounded numbers, used to check the exact one. |
Every written method for adding, subtracting, multiplying and dividing works for one reason: each digit is worth its place. In 3,528 the 5 is worth 500, and in 2.46 the 6 is worth 6 hundredths. The methods keep digits of the same worth together.
For decimals there is one extra idea. Multiplying or dividing by 10 or 100 only moves digits to a new place, so you can work with whole numbers and move the decimal point at the end. That is why $2.4 \times 1.5$ can be worked out as $24 \times 15 = 360$ and then made 100 times smaller: 3.60.
Finally, estimate first. A quick rounded answer catches the place-value slips that careful columns do not.
Another way: picture
Draw $316 \times 24$ as a rectangle 316 long and 24 tall. Cut it across into a strip 20 tall and a strip 4 tall. The big strip is $316 \times 20 = 6320$ square units and the thin one is $316 \times 4 = 1264$. The whole rectangle is the two strips together, 7,584. Each partial product is one strip.
Another way: steps
To multiply 316 by 24, split 24 into 20 and 4. Multiply by the ones first: $316 \times 4 = 1264$. Then multiply by the tens: $316 \times 20 = 6320$. Add the two partial products: $1264 + 6320 = 7584$.
In the column method the second partial product is written one place to the left, with a 0 in the ones place. That 0 is not decoration. It is there because the 2 in 24 means 2 tens, so everything it multiplies is worth ten times as much. Forgetting it gives $1264 + 632 = 1896$, an answer more than four times too small.
An estimate protects you: $300 \times 25 = 7500$, so 7,584 is believable and 1,896 is not.
Long division shares a big number one place at a time, starting with the biggest place. Take $3528 \div 42$.
Divide. 42 does not go into 35, so look at the first three digits, 352. Since $42 \times 8 = 336$ and $42 \times 9 = 378$, 42 goes into 352 eight times. Write 8 above the 2; it is the tens digit of the quotient.
Multiply and subtract. $42 \times 8 = 336$, and $352 - 336 = 16$. The remainder must be less than 42; if it is not, the digit was too small.
Bring down. Bring down the 8 from the ones place to make 168. The 16 was 16 tens, and 16 tens and 8 ones are 168 ones.
Divide again. $168 \div 42 = 4$, with nothing left. Write 4 in the ones place. The quotient is 84.
Rounding the divisor helps you guess each digit. 42 is about 40, and 352 is about 350, so the digit is about $350 \div 40$, which is between 8 and 9. Try 8 first.
One trap: when a divisor does not go into a step, write a 0 in the quotient. $4284 \div 42$: 42 goes into 42 once, then 42 does not go into 8, so write 0, then bring down the 4 to make 84, which is 2 lots of 42. The quotient is 102, not 12.
To add or subtract decimals, line up the decimal points. Write zeros so that both numbers have the same number of places: $6.4 - 2.37$ becomes $6.40 - 2.37 = 4.03$. The zero changes nothing about the value; it only keeps the columns straight.
To multiply decimals, do not line up the points. Multiply the numbers as if the points were not there, then count the decimal places in the two factors and put that many back. For $2.4 \times 1.5$: $24 \times 15 = 360$; there is one place in each factor, two in all, so the answer is 3.60, or 3.6. This works because 2.4 is 24 tenths and 1.5 is 15 tenths, and tenths times tenths are hundredths.
Dividing by a whole number is easy even when the dividend is a decimal: divide as usual and put the point in the quotient straight above the point in the dividend. $25.92 \div 8 = 3.24$.
Dividing by a decimal needs one more move. Multiply the dividend and the divisor by the same power of 10, so that the divisor becomes a whole number. $14.4 \div 0.32$ becomes $1440 \div 32$ after multiplying both by 100. The quotient does not change, for the same reason that $\frac{6}{3}$ and $\frac{60}{30}$ are both 2: scaling both parts of a division by the same number keeps it the same. Then $1440 \div 32 = 45$.
Notice that dividing by a number less than 1 gives an answer bigger than the dividend: there are 45 pieces of 0.32 in 14.4. That is a useful check.
Whatever the operation, the same routine keeps the work reliable.
Step 1: Estimate. Round each number to one or two non-zero digits and work the easy version: $3528 \div 42$ is about $3600 \div 40 = 90$.
Step 2: Set up by place value. Line up decimal points for adding and subtracting. For dividing by a decimal, move both points first.
Step 3: Carry out the method. Partial products for multiplying; divide, multiply, subtract, bring down for dividing.
Step 4: Place the decimal point. For multiplying, count the decimal places in the factors. For dividing, put it above the point in the dividend.
Step 5: Check.
If a check fails, look first at place value: a missing 0 in a partial product, a missing 0 in a quotient, or a decimal point in the wrong place.
In baseball, a batting average is the number of hits divided by the number of official at-bats, written to three decimal places. In 1941 Ted Williams of the Boston Red Sox had 185 hits in 456 at-bats. Long division gives
$$185 \div 456 = 0.4057\ldots$$
which rounds to 0.406. He was the last major league player to bat over 0.400 for a whole season.
Long division works here even though 456 does not go into 185: write 185 as 185.000 and keep bringing down zeros. 456 goes into 1850 four times ($456 \times 4 = 1824$), leaving 26; into 260 zero times; into 2600 five times ($456 \times 5 = 2280$), leaving 320; and into 3200 seven times. The digits 4, 0, 5, 7 give 0.4057. Notice the 0 in the quotient: forgetting it would give 0.457, a very different season.
A class of 28 students takes a bus to a science museum. The bus costs 546 dollars and each museum ticket costs 12.75 dollars. How much should each student pay?
The bus is shared: $546 \div 28 = 19.5$, so 19.50 dollars each. Long division: 28 goes into 54 once, leaving 26; bring down the 6 to make 266, and 28 goes in 9 times ($28 \times 9 = 252$), leaving 14. Write a decimal point and a 0: 28 goes into 140 exactly 5 times.
Each student also pays their own ticket, so the total per student is $19.50 + 12.75 = 32.25$ dollars, with the points lined up. Check: $28 \times 32.25 = 903$ dollars, and the bus plus 28 tickets is $546 + 28 \times 12.75 = 546 + 357 = 903$ dollars. Both ways agree.
Lining up the last digits instead of the points. $5.3 + 2.46$ is not 2.99. Written as $5.30 + 2.46$ with the points in line, it is 7.76.
Dropping the 0 in a partial product. In $316 \times 24$ the second partial product is 6,320, not 632, because the 2 means 2 tens.
Leaving out a 0 in the quotient. $4284 \div 42$ is 102. When 42 does not go into 8, a 0 must be written, or the answer shrinks to 12.
Lining up points to multiply. Decimal points are lined up to add and subtract, never to multiply. For multiplying, count decimal places instead.
Moving the point in only one number. To divide by 0.32, both the divisor and the dividend are multiplied by 100. Moving only the divisor's point changes the question.
Skipping the estimate. Every one of these mistakes changes the size of the answer, and a ten-second estimate catches it.
Work out $316 \times 24$. Estimate first.
$300 \times 25 = 7500$
Rounded numbers give a target, so a place-value slip will stand out.
Multiply by the ones digit, 4.
$316 \times 4 = 1264$
The 4 in 24 is worth 4 ones.
Multiply by the tens digit, which is worth 20.
$316 \times 20 = 6320$
The 2 in 24 is worth 2 tens, so this partial product ends in a 0.
Add the two partial products.
$1264 + 6320 = 7584$
20 lots of 316 and 4 lots of 316 make 24 lots of 316.
Compare with the estimate.
$7584 \approx 7500$
The exact answer is close to the estimate, so no place was lost.
Work out $3528 \div 42$. Estimate first.
$3600 \div 40 = 90$
The quotient should be a two-digit number near 90.
42 does not go into 35, so divide 42 into the first three digits, 352.
$42 \times 8 = 336, \qquad 42 \times 9 = 378$
378 is too big, so 42 goes into 352 eight times. The 8 is the tens digit.
Subtract to find the remainder.
$352 - 336 = 16$
16 is less than 42, which confirms that 8 was the right digit.
Bring down the last digit, 8.
$16 \to 168$
The 16 is 16 tens; with the 8 ones it makes 168 ones.
Divide 42 into 168.
$42 \times 4 = 168$
It goes exactly 4 times, so the ones digit is 4 and there is no remainder.
Read the quotient and check it by multiplying.
$3528 \div 42 = 84, \qquad 84 \times 42 = 3528$
Multiplication undoes division, and 84 is close to the estimate of 90.
A ribbon 14.4 feet long is cut into pieces 0.32 foot long. How many pieces? Write the division.
$14.4 \div 0.32$
The question asks how many groups of 0.32 fit into 14.4.
Estimate the number of pieces.
$14 \div 0.3 \approx 14 \times 3 = 42$
Dividing by about a third is the same as multiplying by about 3, so expect roughly 40 to 50.
Multiply both numbers by 100 to make the divisor whole.
$14.4 \times 100 = 1440, \qquad 0.32 \times 100 = 32$
Scaling the dividend and divisor by the same number leaves the quotient unchanged.
Divide 32 into the first three digits, 144.
$32 \times 4 = 128, \qquad 144 - 128 = 16$
$32 \times 5 = 160$ is too big, so the tens digit is 4 and 16 is left.
Bring down the 0 and divide again.
$160 \div 32 = 5$
16 tens and 0 ones are 160 ones, and $32 \times 5 = 160$ exactly.
Read the quotient and compare it with the estimate.
$1440 \div 32 = 45 \approx 42$
45 pieces is close to the estimate, and it is more than 14.4, as dividing by a number under 1 must give.
Check by multiplying back.
$45 \times 0.32 = 14.40$
45 pieces of 0.32 foot make the whole ribbon: $45 \times 32 = 1440$, with two decimal places.
Write 6.4 with two decimal places and line up the points.
$6.40 - 2.37$
A zero in the hundredths place keeps the columns straight without changing the value.
Subtract the hundredths column, then the rest.
$6.40 - 2.37 = 4.03$
Regroup 1 tenth as 10 hundredths to take away the 7 hundredths.
Multiply by 3.
Multiply: 141 × 88.
The product is answer.
Complete the long division of 2604 by 28.
28 does not go into the first two digits, so divide it into the first three digits, 260. Write the tens digit of the quotient.
$260 \div 28 \rightarrow$ tens
The largest number of 28s that fits into 260 without going over.
Multiply the divisor by that digit and subtract.
$260 - 252 =$ rem
The remainder must be smaller than 28, or the digit was too small.
Bring down the last digit, 4, next to the remainder.
$(\text{remainder}) \times 10 + 4 =$ down
The remainder is in tens, so writing the ones digit beside it makes a number of ones.
Divide 28 into the new number to get the ones digit.
$(\text{new number}) \div 28 =$ ones
It divides exactly, so there is no remainder at the end.
Read the quotient and check it.
$\text{quotient} =$ quot, $\quad 28 \times (\text{quotient}) = 2604$
Multiplying the quotient by the divisor must give back the dividend.
Add: 4.06 + 0.2.
Answer:
Divide: 1222 ÷ 47.
The quotient is answer.
Divide: 38.24 ÷ 8.
The dividend is hundredths hundredths, and the quotient is answer.
A rope is 14.85 feet long. It is cut into pieces that are each 0.45 foot long. How many pieces are there?
Counted in hundredths of a foot, the rope is rope long, so there are pieces pieces.
A rectangular garden bed is 8.8 feet long and 8.6 feet wide. What is its area in square feet?
Without the decimal points the product is whole, so the area is area square feet.
Lesson test: one question per skill, one attempt each, no hints. Your answers are checked when you submit.
A rope is 15.95 feet long. It is cut into pieces that are each 0.55 foot long. How many pieces are there?
Counted in hundredths of a foot, the rope is rope long, so there are pieces pieces.
You can calculate accurately with multi-digit numbers and decimals. Without looking: work out 4.8 divided by 0.6, and say where the decimal point goes and why.
16. Your turn: work out $6.4 - 2.37$, then multiply the answer by 3, step 3
$4.03 \times 3 = 12.09$
$403 \times 3 = 1209$, and the answer keeps the two decimal places of 4.03.