GED Mathematical Reasoning: Understanding Decimals

Decimals are used to describe part of a whole. You’ve probably seen decimals used in situations involving measurement, temperature, and of course, money.

In the decimal system, a unit is divided into ten equal parts. For example, one unit of 100 can be divided into 10 tens. One unit of 10 can be divided into 10 ones. One unit of 1 can be divided into 10 tenths. One unit of 1 tenth can be divided into 10 hundredths, and so on.

Since money is something we all tend to be very familiar with, let’s relate this to dollars and cents.

  • A one hundred dollar bill can be divided equally into 10 ten dollar bills.
  • A ten dollar bill can be divided equally into 10 one dollar bills.
  • A one dollar bill can be divided equally into 10 dimes (where a dime equals one-tenth of a dollar); or 100 pennies (where a penny is equal to one-hundredth of a dollar)

A decimal point is used to separate the whole from the part and we place the representation of the “part” to the right of the decimal point. For instance when we write $2.45, the decimal point separates two whole dollars from the part of a dollar, which is forty-five cents. Note that the “part” of forty-five cents is written to the right of the decimal point and represents 4 dimes (or 4 tenths of a dollar) and 5 pennies (or 5 hundredths of a dollar).

You may be familiar with the idea of writing the value of a coin as the decimal part of a dollar. For examples:

  • One quarter can be written as $0.25 which represents 25 parts of 100
  • One nickel can be written as $0.05 which represents 5 parts of 100
  • One quarter can be written as $0.25 which represents 25 parts of 100
  • One nickel can be written as $0.05 which represents 5 parts of 100

The first three decimal place values to the right of the decimal point and reading from left to right are the tenths place, then hundredths place, then thousandths place. Here is a visual display of the place values:

Whole Number Place Values Decimal Point Decimal Part Place Values
hundreds tens ones  . tenths hundredths thousandths

As a final note, a leading zero is sometimes written to the left of the decimal point as a place holder if there is no whole part. We see this when representing a quarter using the decimal: 0.25

 

Place holder zero

The digit zero has no value, but can be used as a necessary place holder. We can write a zero to the left of the decimal point as a place holder when there is no whole number. For example, when we wrote twenty five cents as 0.25. So, zeros can be added to the far left of the decimal and any whole number portion without changing the value of the decimal. For example: 2.15 is the same as 02.15.

Also, zeros can be added to the far right of a decimal without changing its value. Adding a zero in this way may change how the decimal is read, but it doesn’t change the value. For instance, 0.5 and 0.50 have the same value, although they are read differently.

0.5 is read as “five tenths” (which we can think of as being five dimes)

Whereas

0.50 is read as “fifty hundredths” (which we can think of as being fifty pennies)

We know that five dimes and fifty pennies have the same value, they just look a bit different.

 

Example 1

How is the number 0.025 read in words?

  1. twenty-five thousandths
  2. twenty-five hundredths
  3. twenty-five tenths
  4. point two five thousandths

 

To read a decimal in words, read the number to the right of the decimal point as if it were a whole number and then say the place value of the right-most digit. So we’ll read: zero point zero two five as “twenty five” and add the word “thousandths” since the right most digit – the 5 – is in the thousandths place.

The answer, then, is answer (1) – twenty five thousandths.

When there’s a whole number written to the left of the decimal point, we read the decimal point using the word “and”. For example, you would read the decimal 1.7 as “one and seven tenths”.

Once you are familiar with decimal place value and how to read and write decimals, I may not always use the formal terminology when reading decimals. For example, I may read the decimal 0.24 as “zero point two four” instead of “twenty-four hundredths”.

 

Example 2

How is the following number written using digits? eighty-two hundredths

  1. eight two zero
  2. eight point two zero
  3. eight two point zero
  4. zero point eight two

 

To answer this question requires sort of the opposite process we used in example 1.We will first identify the place value of the right-most digit.

Given “eighty two hundredths”, the right-most digit is in the hundredths place. Now, let’s write the number so that the right-most digit – the 2 – is in the hundredths place, using zeros as place holders if needed. So we’ll write:  zero point eight two, which corresponds to answer (4).

For fun, what if the words given had been eighty-two thousandths?

Well, we would first identify the place value of the right-most digit, which would be the thousandths place. Then, we would write the number so that the right-most digit – the 2 –  is in its proper place, using a zero as a place holder. So we’d write:  zero point ZERO eight two. Notice how we use the zero in the tenths place as a place holder to ensure that the 2 lands in the thousandths spot.

 

Example 3

Put the following decimals in order from least to greatest.

(a) two hundred ninety-four thousandths

(b) thirty-one hundredths

(c) three hundred eleven thousandths

(d) two hundred ninety-five thousandths

  1. order: d, b, c, a
  2. order: d, b, a, c
  3. order: a, d, b, c
  4. order: c, a, b, d

 

To compare decimals, add zeros to the right of the last digit so that each decimal has the same number of digits. Then compare the decimals, digit by digit, starting with the tenths place.

Notice that all of our decimals have a zero to the left of the decimal point. And every decimal value except for (b) has three digits to the right of the decimal point. So let’s add a zero to “thirty one hundredths” so that it has the same number of digits as the other decimals.

(a) 0.294

(b) 0.310

(c) 0.311

(d) 0.295

Now we’ll compare the decimals digit by digit starting with the tenths place. a and d both have a 2 in the tenths place, which means they are smaller than numbers b and d, which have a 3 in the tenths place.

Let’s focus on (a) and (d) to determine which is the smallest value. Moving to the hundredths place, (a) and (d) both have a 9, so we still don’t know which number is smaller. Moving to the thousandths place, (a) contains a 4 and (d) contains a 5. This means that (a) is the smallest number and should be listed first. (d) will be listed second.

Now let’s focus on (b) and (c). They both contain a 3 in the tenths place, which doesn’t tell us which number is smaller. Moving to the hundredths place, (b) contains a 1 and (c) contains a 1. Therefore, we still don’t know which is smaller. Now for the thousandths place. (b) contains a zero and (c) contains a 1, which means (b) is smaller than (c) and should be listed third. Option (c), then, is the largest number. For our final answer, the order from least to greatest should read:  (a), (d), (b), then (c) – which corresponds to answer 3.

Before we conclude this video, I would like to note that reading decimals using place value is a very formal way of stating decimal values. As you continue working with decimals and gain a better understanding of their place value, less formal terminology is required. For example, instead of saying “thirty-one hundredths” I will say “zero point three one”.

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