About Addition on Decimal Fractions
What are Decimal Fractions?
A decimal fraction is a fraction that has powers of 10 in its denominator. That means the value of the denominator is 10 raised to a whole number. The powers of 10 are 10, 100, 1000, 10000, and so on. The general form of a fraction is \(\frac{a}{b}\). In the case of decimal fractions, b will always be a power of 10.
Some examples of decimal fractions are \(\frac{5}{10}, \frac{23}{100}\) and \(\frac{1}{1000}\). We can convert fractions into decimal fractions by multiplying the numerator and the denominator by a number in such a way that the denominator becomes a power of 10. For example, \(\frac{3}{5} = \frac{3 \times 2}{5 \times 2} = \frac{6}{10}\).
A decimal fraction with 10 in the denominator is referred to as tenths. For example, \(\frac{5}{10}\) is 5 tenths and \(\frac{3}{10}\) is 3 tenths. A decimal fraction with 100 in the denominator is referred to as hundredths; \(\frac{14}{100}\) is 14 hundredths and \(\frac{19}{100}\) is 19 hundredths. Similarly, a decimal fraction with 1000 in its denominator is referred to as thousandths; \(\frac{21}{1000}\) is 2 thousandths. Decimal fractions are often used to measure lengths. A meter is \(\frac{1}{1000}^{\text{th}}\) of a kilometer and a centimeter is \(\frac{1}{100}^{\text{th}}\) of a meter.
What are Decimal Numbers?
Decimal numbers are a different form of decimal fractions. We can express any decimal fraction as a decimal number by separating the whole number part and the fractional part by a decimal point. Another way to understand is that the decimal point separates the unit’s place and the tens place. We know that 3 is a decimal fraction equivalent to 3 and 14 hundredths. This means 3 will be in one’s place, 1 in tenths place and 4 in hundredths place. The decimal number equivalent to this fraction is 3.14.
The whole number part and the fractional part are obtained by performing the decimal long division operation on a fraction. Decimal numbers are used in everyday life. The price of goods and the physical quantities like length and weight are often expressed using decimal numbers.
What is Addition of Decimal Fractions?
We are familiar with the addition of fractions having the same denominator. In such cases, we just need to add the numerator of the two addends. Similarly, we can add two decimal fractions having the same denominator by adding the numerators straightaway.
For example, \(\frac{3}{10} + \frac{5}{10} = \frac{3 + 5}{10} =\frac{8}{10}\).
We need to follow these two steps to add decimal fractions with different denominators:
Step 1: Fractions to be added need to have the same denominators. Let’s use equivalent fractions to do that,
For example, to add \(\frac{8}{10}+\frac{14}{100}\), we need to write the equivalent fraction of \(\frac{8}{10}\) which has 100 as its denominator.
\(\frac{8}{10} + \frac{14}{100} = \frac{8 \times 10}{10 \times 10} +\frac{14}{100} = \frac{80}{100}+\frac{14}{100}\)
Step 2: Add the numerators.
\(\frac{80 + 14}{100} = \frac{94}{100}\)
Addition of Decimal Numbers
The addition of decimal numbers is almost the same as the addition of decimal fractions. But there are two additional steps:
Step 1: Write the decimal numbers as fractions.
For example, if you want to find 0.6 + 0.15, convert 0.6 and 0.15 into fractional numbers.
0.6 is 6 tenths. So, \(0.6 = \frac{6}{10}\)
0.15 is 15 hundredths. So, \(0.15 = \frac{15}{100}\)
Step 2: Fractions to be added need to have the same denominators. Let’s use equivalent fractions to do that,
\(\frac{6}{10} +\frac{15}{100} = \frac{6 \times 10}{10 \times 10} + \frac{15}{100} = \frac{60}{100} + \frac{15}{100}\)
Step 3: Add the numerators.
\(\frac{60 + 15}{100} = \frac{75}{100}\)
Step 4: Write the sum as a decimal number.
\(\frac{75}{100} = 0.75\)
So, 0.6 + 0.15 = 0.75
Addition on Decimal Fraction Examples
Example 1: Find \(\frac{48}{100} + \frac{4}{10}\).
Solution:
Step 1: Fractions to be added need to have the same denominators. Let’s use equivalent fractions to do that. Here, we will find the equivalent fraction of \(\frac{4}{10}\).
\(\frac{48}{100} + \frac{4}{10} = \frac{48}{100} + \frac{4 \times 10}{10 \times 10} = \frac{48}{100} + \frac{40}{100}\)
Step 2: Add the numerators.
\(\frac{48 + 40}{100} = \frac{88}{100}\)
Therefore, \(\frac{48}{100} + \frac{4}{10} = \frac{88}{100}\)
Example 2: Find 0.309 + 0.41 + 0.1.
Solution:
Step 1: Write the decimal numbers as fractions.
\(0.309 + 0.41 + 0.1 = \frac{309}{1000} + \frac{41}{100} + \frac{1}{10}\)
Step 2: Fractions to be added need to have the same denominators. Let’s use equivalent fractions to do that,
\(\frac{309}{1000} + \frac{41}{100} + \frac{1}{10} = \frac{309}{1000} + \frac{41 \times 10}{100 \times 10} + \frac{1 \times 100}{10 \times 100} = \frac{309}{1000} + \frac{410}{1000} + \frac{100}{1000}\)
Step 3: Add the numerators.
\(\frac{309 + 410 + 100}{1000} + \frac{819}{1000}\)
Step 4: Write the sum as a decimal number.
\(\frac{819}{1000} = 0.819\)
Therefore, 0.309 + 0.41 + 0.1 = 0.819
Example 3:
Andrew has three nickels, a dime, and a rare half-dollar coin. Find the total amount of money using fractions.
Solution:
1 cent is one-hundredth of a dollar. A nickel is worth 5 cents.
Value of 1 nickel \(¢5 = $5 \times \frac{1}{100} = $\frac{5}{100}\)
Value of 3 nickels \( = $3 \times \frac{5}{100} = $\frac{15}{100}\)
Value of a dime is ten cents or a tenth of a dollar.
Value of 1 dime \( = ¢10 = $10 \times \frac{1}{100} = $\frac{1}{10}\)
Value of half-dollar coin \( = $\frac{1}{2}\)
Total amount \( = $\frac{15}{100} + $\frac{1}{10} + $\frac{1}{2}\)
\( = $\frac{15}{100} + $\frac{1 \times 10}{10 \times 10} + $\frac{1 \times 50}{2 \times 50}\)
\( = $\frac{15}{100} + $\frac{10}{100} + $\frac{50}{100}\)
\( = $(\frac{15 + 10 + 50}{100})\)
\( = $\frac{75}{100}\)
Therefore, Andrew has \(\frac{75}{100}\) dollars or 75 cents.
Example 4:
Tina bought three chocolates whose prices are $0.35, $0.25, and $0.2. Find the amount of money she paid for these chocolates.
Solution:
To find the total amount she spent on chocolate, we need to find $0.35 + $0.25 + $0.2 by following these steps.
Step 1: Write the decimal numbers as fractions.
\(0.35 + 0.25 + 0.2 = \frac{35}{100} + \frac{25}{100} + \frac{2}{10}\)
Step 2: Fractions to be added need to have the same denominators. Let’s use equivalent fractions to do that,
\(\frac{35}{100} + \frac{25}{100} + \frac{2}{10} = \frac{35}{100} +\frac{25}{100} + \frac{2 \times 10}{10 \times 10} = \frac{35}{100} + \frac{25}{100} + \frac{20}{100}\)
Step 3: Add the numerators.
\(\frac{35 + 25 + 20}{100 = \frac{80}{100}\)
Step 4: Write the sum as a decimal number.
\(\frac{80}{100} = 0.8\)
Example 5:
Observe the model and write the equivalent decimal number.
Solution:
Let’s express the picture using actual numbers.
\(5 + \frac{3}{10} + \frac{43}{100}\)
Now, let’s add these fractions to get the final sum.
\(5 + \frac{3}{10} + \frac{43}{100} = 5 + \frac{30}{100} + \frac{43}{100}\)
\( = 5 + \frac{30+43}{100}\)
\( = 5 + \frac{73}{100}\)
= 5 + 0.73
= 5.73
Any two fractions that represent the same number are known as equivalent fractions. One among the equivalent fractions can be reduced to get the other fraction by dividing it with a common factor.
For example, in \(\frac{1}{2}\) and \(\frac{2}{4}, \frac{2}{4}\) can be reduced further by dividing by 2 to get \(\frac{1}{2}\), hence \(\frac{1}{2}\) and \(\frac{2}{4}\) are equivalent fractions.
A decimal fraction is a fraction whose denominator is a power of 10 (10, 100, 1000, and so on). A decimal number is a decimal fraction expressed using a decimal point. In a decimal number, the whole number part and the fractional part are separated using a decimal point.