## Decimals and Place Value

Decimals are numbers written using the base-ten place-value system. The chart below shows the names of the place values. Each place value is 10 times the value to its right.

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.
1 = units digit
10 = tens digit
100 = hundreds digit
1,000 = thousands digit

Another example: 34,567

3  is the ten-thousands digit
4  is the thousands digit
5  is the hundreds digit
6  is the tens digit
7  is the units digits

The GMAT often asks questions using digits.

##### Example

Show the value of a number with x as the units digit, y as the tens digit, and z as the hundreds digit.

### Solution

You can use the Experiment strategy as a way to recall place value. Each place value is 10 times the value to its right.
347 = three hundred forty seven
= (3 × 100) + (4 × 10) + (7 × 1)

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 = 100+ 10y + x.

The GMAT can give puzzle problems. Use the strategy of reading the entire question before you start.

##### Example

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?

### Solution

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.

## Rounding

##### Example

Write each fraction as a decimal.

(a) \dfrac{3}{100}

(b) \dfrac{23}{1000}

(c) \dfrac{45}{10}

### Solution

(a) \dfrac{3}{100} = 0.03              …The denominator has 2 zeros, so there are 2 digits to the right of the decimal point. You need to add a zero.

(b) \dfrac{23}{1000} = 0.023       …The denominator has 3 zeros, so there are 3 digits to the right of the decimal point. You need to add a zero.

(c) \dfrac{45}{10} = 4.5                 …The denominator has 1 zero, so there is just 1 digit to the right of the decimal point.

Note: The leading zero, before the decimal point, is there to make the decimal more visible:  0.35 = .35, but the zero makes easier to see the decimal point and not confuse 0.35 with 35.

Decimals can be used to write other fractions. Use the fraction bar as a division symbol.

##### Example

Write each fraction as a decimal.

(a) \dfrac{3}{8}

(b) \dfrac{1}{3}

### Solution

(a) \dfrac{3}{8} = 0.375              …This is a terminating decimal. The division stops.

(b) \dfrac{1}{3} = 0.333… = 0.3  …This is a repeating decimal. The division repeats without end.

Decimals can be written as fractions. Remember to simplify the fractions.

##### Example

Write each decimal as a fraction.

(a) 0.3

(b) 0.075

(c) 31.2

### Solution

(a) 0.3 = \dfrac{3}{10}

(b) 0.075 = \dfrac{75}{1000} = \dfrac{3}{40}

There are 3 digits to the right of the decimal point, so the denominator under 75 has 3 zeros.

(c) 31.2 = 31\dfrac{\,2\,}{10} = 31\dfrac{\,1\,}{5}

The answer could also be an improper fraction.

31.2 = 31\dfrac{\,1\,}{5} = \dfrac{156}{5}

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.

##### Example

Add:  5 + 2.783 + 3.04

### Solution

 5.000 2.783 \dfrac{+\,3.040}{10.823} …Add zeros as placeholders.
##### Example

Calculate:
6.98 + 3.217 – 3.637 + 4

### Solution

The simple way to solve this problem is to do the addition then the subtraction.

Find the sum.

6.980
3.217
\dfrac{+\,4.000}{14.197}

Subtract.

14.197
\dfrac{-\,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.

(a) 6 × 3.02

(b) 0.4 × 0.06

### Solution

(a)  Multiply as integers.
× 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

(b)  Multiply as integers.
× 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.

##### Example

Divide:  \dfrac{42.75}{9.5}

### Solution

Move the decimal point in 9.5 one place to the right to get the whole number 95. Then move the decimal point in 42.75 one place, to 427.5.

——————————————–4.5
9.5)\overline{\,42.75} = 95)\overline{\,427.5}  =  95)\overline{\,427.5}

Keep the decimal points aligned in the dividend and the quotient.

You can check the decimal point placement in your answer by looking at the multiplication.
4.5 × 9.5 will have 2 decimal places in the answer, and 42.75 has 2 decimal places.

##### Example

Divide:  \dfrac{50}{0.02}

### Solution

Move the decimal point in 0.02 two decimal places to get the whole number 2. To move the decimal place in 50 by 2 places, you need to add 2 zeros to get 5,000.

————————————2500
—-0.02)\overline{\,50} = 2)\overline{\,5000} = 2)\overline{\,5,000}

Keep the place values aligned; the answer is 2,500, not 25.