QuanticaPrealgebraOpen in the courseDecimal arithmetic is not new arithmetic. Every digit still sits in a place worth ten times the place to its right, and a decimal is just a fraction over a power of ten. The only genuinely new job is bookkeeping, knowing exactly where the point belongs in the answer. Adding, multiplying, and dividing each handle that job differently, so we will work through all three.
Problem
A runner covers km then km. Stack the numbers with decimal points aligned, pad to , and add column by column. What is in km?
Show a hint
- Write the two numbers one above the other with the decimal points lined up. Since has no hundredths digit, fill that empty spot with a trailing zero so it reads , which is the same value.
- Add the columns from the right. Hundredths give , tenths give so write and carry , and ones give . Bring the decimal point straight down.
Show the full solution
Stack the numbers with the points aligned and pad to so both show a hundredths digit. Adding right to left, hundredths give , tenths give so write and carry , and ones give . The point drops straight down. The runner covers km.
Problem
A café lists a sandwich at dollars and a cookie at dollars. Align the decimal points, pad to , and add. What is the correct total?
Show a hint
- A decimal place only means something next to its neighbors. The tenths digit must sit above the tenths digit, and hundredths above hundredths. Stack the numbers so the two decimal points fall in one straight vertical line, not so the last digits line up.
- Write on top and underneath with the points aligned. Add the hundredths, . Add the tenths, . Bring down the and drop the point straight down.
Show the full solution
Pad to so both numbers show two decimal places, then stack them with the points in one vertical line. The hundredths give , the tenths give , and the comes straight down. The total is dollars. Line up the right edges instead and the tenths sits above the hundredths, adding digits of different sizes and inflating a few cents into a few dollars.
Problem
A bottle holds litres. After pouring, litres remain. Line up the decimal points and subtract. How many litres were poured out?
Show a hint
- Write as first. Now both numbers have a tenths digit and a hundredths digit, so every column has something to subtract.
- In the hundredths column needs a borrow, so take tenth, making it . In the tenths column the became , so borrows again, giving . In the ones column . Read it off with the point in place.
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Pad to and stack it over with the points aligned. In the hundredths column, borrow tenth to get . That leaves in the tenths, so borrow one to get . The ones column is then . Bringing the point straight down gives litres. Checking it, , the full bottle again.
Problem
Your gut might say . Resist it and compute instead. What is ?
Show a hint
- Turn each decimal into a fraction over ten. You are multiplying by , so multiply the tops together and the bottoms together.
- The tops give and the bottoms give , so you have . A denominator of means two digits after the point, and you need a leading zero to fill the empty tenths place.
Show the full solution
Write each decimal as a fraction over ten and multiply across, . A denominator of puts the digits two places past the point, so the answer is and not . So . Taking a fraction of a fraction always lands below both factors, which is why the product is smaller than and .
Problem
A herb bed measures m by m. Write , , multiply as fractions, and read off the decimal area. What is in square metres?
Show a hint
- You already did the hard part above. The fractions multiplied to , so now just turn that back into a decimal.
- is forty-two hundredths. Hundredths live two places after the point, so write and slide the point so the last digit lands in the second place: .
Show the full solution
Write the decimals as fractions and multiply across, . A denominator of is the hundredths place, so that reads as forty-two hundredths, and the bed is square metres. There is a shortcut hiding in that. The digits gave , and each factor had one decimal place, so the product carries two.
Problem
Almonds cost dollars per scoop. You take scoops. Multiply , then place the decimal point. What do you owe in dollars?
Show a hint
- A whole number does not break the rule, it just brings zero decimal places to the table. Start by ignoring the point and multiplying the digits as whole numbers, .
- You found . Now count decimal places in both factors together. The has none and has two, so the product needs two places. Put the point two digits from the right of .
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Strip the point and multiply the digits like ordinary whole numbers, . Now total the decimal places in the two factors. The whole number carries zero places and carries two, so the product keeps places. Placing the point two digits from the right turns into . A quick gut check, is a quarter of a dollar, so seven quarters is dollars, exactly as expected. So dollars.
Problem
A recipe needs of a gram pinch of saffron. Multiply: , count three total decimal places, and place the point. How many grams is that?
Show a hint
- Multiply as whole numbers first, , then count the decimal places in both factors. Two places plus one place means the answer needs three places total.
- You have the digit but it must sit three places after the point. Fill the first two places after the point with zeros and put the in the third, the thousandths spot.
Show the full solution
Ignore the points and multiply the digits, . Then count places. has two and has one, so the product needs . Two zeros push the out to the thousandths spot, so the recipe uses grams. The fractions agree, since and thousandths is exactly that third place.
Problem
A m rope is cut into m pieces. Write as , multiply top and bottom by to get , then divide. How many pieces?
Show a hint
- The divisor is , which has one decimal place. Slide the point one place to the right in both numbers so the divisor becomes a whole number. What do and turn into?
- Sliding both points one place gives . Now it is plain whole-number division, and goes into exactly how many times?
Show the full solution
Write the division as and multiply the top and bottom by , which slides each point one place right and gives . Since , the rope makes pieces. Multiplying top and bottom by the same number is multiplying by , so the slide cannot change the value.
Problem
Find . Write it as , multiply top and bottom by to clear the divisor, then divide. What is the quotient?
Show a hint
- Multiply both and by so the divisor turns whole. That leaves you with , which has the same value but is much friendlier.
- Divide by . Since goes into once with left, and goes into exactly times, the quotient is . Bring the decimal point straight up and you land on .
Show the full solution
Multiply the top and bottom of by to clear the divisor, which turns the problem into . Twelve fits into once, leaving , and twelve fits into exactly times, so . Check it by multiplying back, since .
Problem
A maker has spools of m wire each and uses m. Find the total wire on hand, then subtract what is used. How many metres are left?
Show a hint
- Two clean steps. Step one is a whole number times a decimal, so multiply to get the total length on hand. Step two is a subtraction, the total minus the m used.
- Multiply , and since has one decimal place the total carries one place too, giving m. Now line up the points and subtract , padding to so both numbers show hundredths.
Show the full solution
Three spools of m each is . The digits give , and carries one decimal place, so there is m on hand. Pad that to , line up the points, and subtract what the build uses. That leaves m of wire.
Practice these ideas
Practice
Two reef stretches measure m and m. Line up the decimal points and add. What is ?
Show the solution
Both numbers already show one decimal place, so stack them with the points aligned and add. The tenths give and the ones give , with the point dropping straight down. The two stretches total meters.
Practice
A stack holds liters. You pour out liters. Pad to and subtract. How many liters remain?
Show the solution
Pad to so both numbers show hundredths, then line up the points and subtract from the right. The hundredths borrow to give . The tenths are now , so borrow again for . The ones give . Bringing the point down leaves liters.
Practice
A drone starts with units of battery and the trip uses units. Write as and subtract. How many units remain?
Show the solution
Write as , the same value with two decimal places, and stack it over with the points aligned. The hundredths borrow to give , the tenths become as the borrow cascades, and the ones give . So units of charge are left. Checking it, , a full battery again.
Practice
A square sticker measures cm by cm. Multiply the digits, count two total decimal places, and place the point. What is the area?
Show the solution
Ignore the decimal points and multiply the digits like whole numbers, . Then count the decimal places in the factors, one in and one in , which makes two in all. Giving two decimal places turns it into .
Practice
Fog covers of a window. By noon it blocks only of that area. What fraction of the whole window is still fogged? Give your answer as a decimal.
Show the solution
Half of the fogged area means . The digits give , and each factor has one decimal place, so the product needs two, giving of the window. It landed below both and , which is what taking a part of a part always does.
Practice
A vial holds grams of saffron. A recipe uses of the vial. What is in grams?
Show the solution
Set the points aside and multiply like whole numbers, . There is one decimal place in and one in , so the product needs two, and counting two spots from the right of gives . Trailing zeros after the point add nothing, so the recipe uses gram. Two numbers that both carry decimals can still multiply to a clean whole number, when the places fill in exactly.
Practice
A reading is grams. Another mass is of that first amount. What is ?
Show the solution
Start with the digits and forget the decimals. Multiplying gives . Now count how many decimal places the factors have. The number has two places and has one place, for a total of three. The product must therefore have three decimal places. Writing with three places means filling in zeros, so sits in the thousandths spot as .
Practice
Slide both numbers in one place right to make the divisor whole. What is ?
Show the solution
Sliding both numbers one place to the right means multiplying each by , and doing the same thing to both keeps the quotient the same. So becomes . Now the divisor is a whole number and the work is easy. Since , we get . Therefore .
Practice
A ribbon is m long. A pattern piece needs m. How many times as long as one strip is the whole ribbon? Find .
Show the solution
Start by making the divisor whole. Slide both points one place to the right, so becomes and becomes . The quotient does not change, so . Now divide. goes into once, leaving . Bring down the to make , and , so the next digit is . Keeping the decimal point in place gives . Check by multiplying back, , which matches the ribbon. So the ribbon is times as long as one strip.
Practice
Add mm and mm. Pad the shorter number so both show two decimal places, then add. What is ?
Show the solution
Pad to and line up the points. The hundredths give and the tenths give . The total is mm. The inner zero in is doing real work, since it holds the in the hundredths place instead of letting it drift up to the tenths.
Practice
A kit needs half of a m wire length. Multiply . Count decimal places and place the point. What is the result?
Show the solution
Ignore the points and multiply the digits, . There is one decimal place in and one in , so the answer needs two, giving m. That matches taking half of .
Practice
You buy jars of jam at dollars each and pay with a dollar bill. Find the total cost, then subtract from . How much change do you get?
Show the solution
Start with the cost of the jam. Multiply . Ignoring the point for a moment, . The factor has two digits after the point, so the product does too, giving . Now find the change. Write the dollar bill as so the points line up, then subtract. . So your change is dollars.