Decimals are numbers written using the base-ten place-value system. A very visible use of decimals is money. The chart below shows the names of the place values. Each place value is 10 times the value to its right. The decimal point is usually read as “and.”
1,004,307.6 is one million, four thousand, three hundred seven and six tenths
60.543 is sixty and five hundred forty three thousandths
Each digit of a number has a name based on place value.
Since z is the hundreds digit, y is the tens digit, and x is the units digit, the number is zyx. The value is zyx = 100z + 10y + x.
The GMAT can give puzzle problems. Use the strategy of reading the entire question before you start.
A number has two digits to the left of the decimal point and two digits to the right of the decimal point. The hundredths digit is two times the tenths digit. The number has 2 as the tenths digit. The ones digit is greater than zero and less than the tenths digit. When 1 is subtracted from the tens digit, the result is 5. What is the number?
Draw a quick picture of the number. ____ ____ . ____ ____
A sentence in the middle of the question starts the process: the tenths’ digit is 2.
The next sentence uses the ones digit, so find that. The ones digit is the only whole number between 0 and 2, so is 1.
The tens digit minus 1 equals 5, so the tens digit is 6.
Then back to the second sentence. The hundreds digit is two times the tenths digit, 2 × 2 = 4.
So the answer is 61.24.
Rounding can be used to get a value that’s easier to use in calculations, or to get an estimate for the answer. For example, if you are buying a shirt for $19.99 and pants for $28.49, a quick estimate of the total is $20 + $28 = $48.
To round a decimal to a given place value, look at the digit in the place to the right.
If the digit is less than 5, round down.
If the digit is 5 or greater, round up.
Using a number line helps visualize rounding. Round to the value closest to the number.
Round each number to the nearest unit and nearest tenth.
6.25 is closer to 6 than 7, so it rounds to 6 as the nearest unit. Since the digit to the right is 5, 6.25 rounds to 6.3 as the nearest tenth.
Since the tenths digit is 5, 8.5 rounds up to 9, and remains 8.5 for the tenths.
9.846 rounds to 10 and to 9.8. Do not round up from 9.846 to 9.85 before rounding to the tenths’ place. (If you did you would get 9.9, which is incorrect.)
Decimals and Fractions
Decimals can be used to write fractions. If the fraction has a denominator that is a multiple of ten, and the numerator does not end in 0, then the number of zeros in the denominator will be the number of places to the right of the decimal.
0.075 = 75/1000 = 3/40. There are 3 digits to the right of the decimal point, so the denominator under 75 has 3 zeros.
31.2 = 312/10 = 311/5 The answer could also be an improper fraction. 31.2 = 311/5 = 156/5
Adding and Subtracting Decimals
To add or subtract decimals, use a vertical format and line up the decimals. You can add trailing zeros as placeholders at the end of a decimal to make the alignment easier. Be sure to write the decimal point in the answer.
The simple way to solve this problem is to do the addition then the subtraction.
Find the sum. _ 6.980 _ 3.217 + 4.000 14.197
Subtract. 14.197 – 3.637 10.560
Multiplying and Dividing Decimals
Multiply two decimals just like you would multiply two integers, then place the decimal point in the answer. If both factors have decimals, the number of decimal places in the product is equal to the sum of number of the decimal places in the two factors.
Multiply as integers. 6 × 302 = 1,812 There are 2 decimal places in 3.02 and none in 6, so there will be 2 decimal places in the product. 6 × 3.02 = 18.12 You can check your answer using rounding. 6 × 3 = 18, so your answer is reasonable.
Multiply as integers. 4 × 6 = 24 There is 1 decimal place in 0.4 and 2 decimal places in 0.06, so count in 3 decimal places from the right. This means you need to add a zero after the decimal point, to have the 3 decimal places. 0.4 × 0.06 = 0.024
To divide decimals, move the decimal point in the divisor to the right so it’s a whole number. Then move the decimal point in the dividend the same number of places.