Exercises: decimal numbers (mixed)

1) Convert $9.56$ to a fraction: $9.56=9\frac{56}{100}=\frac{956}{100}$.

2) $\frac{3}{4}$ and $\frac{956}{100}$ are unlike fractions. The LCM of 4 and 100 is 100.

3) $\frac{3}{4}$ only has to be written as an equal fraction with denominator 100 by multiplying and dividing:

$\frac{3}{4}=\frac{3\cdot 25}{4\cdot 25}=\frac{75}{100}$.

4) Find the sum: $\frac{3}{4}+9.56=\frac{75}{100}+\frac{956}{100}=\frac{1031}{100}$.

5) $\frac{3}{4}+9.56=\frac{1031}{100}=10\frac{31}{100}=10.31$

If a power of 10 is a multiple of the denominator, there is another way to find the sum of a decimal and a fraction. It is as follows:

1. Convert the fraction to a decimal.

2. Add the two decimals.

$\frac{3}{4}+9.56$:

1) Convert $\frac{3}{4}$ to a decimal:

100 is a multiple of 4. By multiplying and dividing with 25 $\frac{3}{4}$ can be converted to an equal fraction:

$\frac{3}{4}=\frac{3\cdot 25}{4\cdot 25}=\frac{75}{100}=0.75$

2) Find the sum: $\frac{3}{4}+9.56=0.75+9.56=10.31$

You need an adequate knowledge of repeating decimals, of converting a decimal to a fraction and of adding fractions to find the answer.

$\frac{2}{3}$ converts to a repeating decimal. Hence, finding the sum is easier if you convert the terminating decimal and the integer to a fraction with a common denominator.

Adhering to the following steps will give you the answer.

1. Convert the terminating decimal and the integer to a fraction with a common denominator.

2. Find the least common multiple of all three denominators. The LCM will be the common denominator of the three fractions.

3. Multiply and divide the fractions if necessary.

4. Add the fractions.

5. Simplify.

1) $1.39=\frac{139}{100}$, $4=\frac{400}{100}$

2) The LCM of 3 and 100 is 300.

3) Each of the three fractions has to be converted to an equal fraction with denominator 300:

$\frac{2}{3}=\frac{2\cdot 100}{3\cdot 100}=\frac{200}{300}$ , $\frac{400}{100}=\frac{400\cdot 3}{3\cdot 100}=\frac{1200}{300}$, $\frac{139}{100}=\frac{417}{300}$

4) Add the fractions.

$\frac{2}{3}+4+1.39=\frac{200}{300}+\frac{1200}{300}+\frac{417}{300}=\frac{1817}{300}=\frac{1800}{300}+\frac{17}{300}=6\frac{17}{300}$

5) $\frac{17}{300}$ cannot be simplified. Hence, the answer is $6\frac{17}{300}$.

For answering this problem, you must have adequate knowledge of repeating decimals, of converting decimals to fractions and of adding fractions.

As $\frac{2}{7}$ converts to a repeating decimal, finding the sum will be easier if the terminating decimal is converted to a fraction.

Adhering to the following steps will give you the answer.

1. Convert the terminating decimal to a fraction.

2. Find the least common multiple of the two denominators. The LCM will be the common denominator of the two fractions.

3. With the LMC as the common denominator for the two fractions, multiply and divide each fraction.

4. Add the two fractions.

5. Simplify the sum if possible.

So, Let's get started.

1) $0.4=\frac{4}{10}=\frac{2}{5}$ .

2) $\frac{2}{7};\frac{2}{5}$: The LCM of denominators 5 and 7 is $5\cdot 7=35$.

3) Multiply and divide each fraction to convert it to an equal fraction.

$\frac{2}{5}=\frac{2\cdot 7}{5\cdot 7}=\frac{14}{35}$, $\frac{2}{7}=\frac{2\cdot 5}{7\cdot 5}=\frac{10}{35}$ .

4) Add the fractions.

$0.4+\frac{2}{7}=\frac{14}{35}+\frac{10}{35}=\frac{24}{35}$.

5) $\frac{24}{35}$ cannot be simplified. Hence the answer is $\frac{24}{35}$.

You need adequate knowledge of repeating decimals, of converting a (terminating) decimal to a fraction and of adding fractions and mixed numbers.

$\frac{1}{7}$ converts to a repeating decimal, $1.7$ is a terminating decimal. Hence, finding the sum $\frac{1}{7}+1.7$ is easier if $1.7$ is converted to a mixed number. Mixed numbers are usually preferred when adding or subtracting improper fractions.

Adhering to the following steps will give you the answer.

1. Convert the decimal to a mixed number.

2. Find the least common multiple of $\frac{1}{7}$ and the denominator of the fraction part of the mixed number. It is the common denominator of $\frac{1}{7}$ and the fraction part of the mixed number.

3. Multiply and divide $\frac{1}{7}$ and the fraction part of the mixed number.

4. Add $\frac{1}{7}$ to the mixed number.

5. Simplify the fraction part if possible.

1) $1.7=1\frac{7}{10}$.

2) The least common multiple of 7 and 10 is $7\cdot 10=70$.

3) Multiplying and dividing:

$\frac{1}{7}=\frac{1\cdot 10}{7\cdot 10}=\frac{10}{70}$, $\frac{7}{10}=\frac{7\cdot 7}{10\cdot 7}=\frac{49}{70}$.

4) $\frac{1}{7}+1.7=\frac{10}{70}+1\frac{49}{70}=1\frac{59}{70}$.

5) The fraction part of $1\frac{59}{70}$ cannot be simplified.

Hence the answer is $1\frac{59}{70}$.

This is an easy problem. All you need to know is how to convert an improper fraction to a decimal and how to add decimals.

When adding or subtracting, using mixed numbers is preferred over improper fractions.

The answer can be found after having carried out three steps.

1. Convert the improper fraction to a mixed number.

2. If the mixed numbers do not have the same number of decimal places, fill any gaps with a zero. Write the decimal numbers in a vertical list, lining up the decimal points.

3. Starting with the column with the least place value, add the digits of the column. If the sum is ten or more, do not forget to add the tens digit to the column sum on the left. Do not forget the decimal point in the answer either. It must be lined up with the other decimal points.

So, let's get started.

1) $\frac{3}{2}=\frac{2+1}{2}=\frac{2}{2}+\frac{1}{2}=1\frac{1}{2}=1\frac{5}{10}=1.5$

2) 1.5 and 3.44 do not have the same number of place values. Fill the gap with a zero:

1.5 = 1.50

Adding the decimals:

$\begin{array}{l}1.50\\ \underline{+3.44}\\ 4.94\end{array}$

So, $\frac{3}{2}+3.44=4.94$ .

1. Convert the decimal to a mixed number. Do not simplify the fractional part as the denominator is 10. Multiples of 10 can be easily calculated.

2. Find the least common multiple of the fractions and the fractional part of the mixed number. The LCM is the common denominator of the the fractions.

3. Multiply and divide each fraction.

4. Using the communtative property of addition, first find the sum of the two fractions. Then add this sum to the mixed number.

5. If possible, simplify the fractional part of the mixed number.

So, let's get started.

1) Convert 1.2 to a mixed number.

$1.2=1\frac{2}{10}$ .

2) Find the LMC of denominators 9, 10, and 7. Clearly, the highest common factor of these numbers is 1. Hence, the LCM of the numbers is the product of 9, 10, and 7.

$9\cdot 10\cdot 7=630$. This is the common denominator.

3) Multiply and divide.

$\frac{4}{9}=\frac{4\cdot 10\cdot 7}{9\cdot 10\cdot 7}=\frac{280}{630}$,

$\frac{2}{10}=\frac{2}{10\cdot 9\cdot 7}=\frac{126}{630}$,

$\frac{2}{7}=\frac{2\cdot 10\cdot 9}{7\cdot 10\cdot 9}=\frac{180}{630}$.

4) Using the commutative law of addition:

$\frac{4}{9}+\frac{2}{7}=\frac{280}{630}+\frac{180}{630}=\frac{460}{630}$

$\frac{460}{630}+1.2=\frac{460}{630}+1\frac{126}{630}=1\frac{586}{630}.$

5) The fractional part $\frac{586}{630}$of $1\frac{586}{630}$ can be simplified to $\frac{293}{315}$.

So, $\frac{4}{9}+1.2+\frac{2}{7}=1\frac{293}{315}$.

This is a fairly easy problem.

Knowing how to add decimals as well as how to convert a fraction to a decimal is a prerequisite for tackling this problem.

$\frac{1}{4}$ and $\frac{3}{8}$ convert to terminating decimals. Hence, the above sum will be rewritten in decimal notation and then added as this is quicker than adding fractions.

Adhering to the following steps will take you to the answer.

1. Convert $\frac{1}{4}$ and $\frac{3}{8}$ to a decimal.

2. Add the decimals using column addition: Make sure all decimals have the same number of decimal places, fill in any gaps with a zero if necessary; do not forget to line up the decimal points; do not forget to add the decimal point in the answer; if the sum of digits of a column is 10 or more do not forget to carry the tens digit to the column to the left.

So let's get started.

1) Convert to a decimal.

$\frac{1}{4}=\frac{1\cdot 25}{4\cdot 25}=\frac{25}{100}=0.25$.

$\frac{3}{8}=\frac{3\cdot 125}{8\cdot 125}=\frac{375}{1000}=0.375$.

2) Add, using column addition.

$\begin{array}{l}0.250\\ 3.200\\ 0.375\\ \underline{+1_1{.7}_{1}50}\\ 5.525\end{array}$

So, $\frac{1}{4}+3.2+\frac{3}{8}+1.7=5.525$.

1) Convert the decimal to a fraction.

$0.3=\frac{3}{10}$.

2) Find the least common multiple.

The denominators are 7, 10 and 5. As 10 is a multiple of 5, it suffices to find the least common multiple of 7 and 10. As the highest common factor of 7 and 10 is 1, the least common multiple (LCM) of 7 and 10 is $7\cdot 10=70$.

3) Multiply and divide.

$\frac{5}{7}=\frac{5\cdot 10}{7\cdot 10}=\frac{50}{70}$.

$\frac{3}{10}=\frac{3\cdot 7}{10\cdot 7}=\frac{21}{70}$.

$\frac{2}{5}=\frac{2\cdot 2\cdot 7}{5\cdot 2\cdot 7}=\frac{28}{70}$.

4) Add the fractions.

$\frac{5}{7}+0.3+\frac{2}{5}=\frac{50}{70}+\frac{21}{70}+\frac{28}{70}=\frac{99}{70}$.

5) Convert the improper fraction to a mixed number, simplify the fractional part.

$\frac{99}{70}=\frac{70+29}{70}=\frac{70}{70}+\frac{29}{70}=1\frac{29}{70}$.

The fractional part of $1\frac{29}{70}$ cannot the simplified.

Done! The answer is $1\frac{29}{70}$.