Exercises: decimal numbers (mixed)

$17.17+0.3$

Use the written addition to add the two decimal fractions. You must insert an additional zero as the second decimal place for the $0.3$.

$\def\arraystretch{1.25} \begin{array}{l}\hphantom{+}17{.}17\\\underline{+\hphantom{1}0{.}30}\\\hphantom{+}17{.}47\end{array}$

Alternative way

Convert the decimal numbers to fractions and bring them to a common denominator.

$18.7-1.87$

Subtract the two decimal fractions using written subtraction. You must insert an additional zero as a decimal place.

$\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}1\overset{7}{\not8}.\overset{\overset{16}{\not6}}{\not{7}}\overset{10}{\not0}\\\underline{-\hphantom{1}\;1.\;8\;7}\\\hphantom{-}1\;6.\;8\;3\end{array}$

Alternative way

Convert into decimal fractions.

$1.2\cdot0.12$

Apply written multiplication to the decimal fractions.

$\def\arraystretch{1.25} \begin{array}{r}\underline{1{.}2\cdot0{.}12}\\24\\\underline{\hphantom{1{.}2\cdot{,}}120}\\0{.}144\end{array}$

Alternative way

Convert into decimal fractions.

$0.8:0.32=80:32$

Multiplying by 100 in the denominator and the enumerator does not change the value of the fraction. Use the written division.

$\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}80:32=2{.}5\\\underline{-64}\\\hphantom{-}160\\\underline{-160}\\\hphantom{-16}0\end{array}$

Alternative way

$0.32:0.6=32:60$

Multiplying by 100 in the denominator and the enumerator does not change the value of the fraction. Use the written division.

$\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}32:60=0{.}533…=0{.}5\overline{3}\\\hphantom{-}320\\\underline{-300}\\\hphantom{-3}200\\\hphantom{3}\underline{-180}\\\hphantom{-11}200\\\hphantom{30}\underline{-180}\\\hphantom{--}200\\\hphantom{-111}\vdots\end{array}$

Alternative way

$0.0123:1000$

When dividing by 1000, the decimal point is shifted by 3 places to the left.

$=0.0000123$

Alternative way

$0.0123\cdot100$

When multiplying by 100, move the decimal point two places to the right.

$=1.23$

Alternative way

$5.5\cdot0.12:0.1$

First, compute $5.5\cdot0.12$.

$\def\arraystretch{1.25} \begin{array}{r}\underline{5{.}5\cdot0{.}12}\\110\\\underline{\hphantom{5{,}5\cdot{,}}550}\\0{.}660\end{array}$

Then, compute $0.66:0.1$.

$0.66:0.1=66:10$

Multiplying by 100 in the denominator and the enumerator does not change the value of the fraction.

$\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}66:10=6{.}6\\\underline{-60}\\\hphantom{-6}60\\\hphantom{1}\underline{-60}\\\hphantom{-60}0\end{array}$

$\Rightarrow5.5\cdot0.12:0.1=6.6$

Use the written method of division.

Alternative way

Alternative way

$\def\arraystretch{1.25} \begin{array}{l}\hphantom{-}65:11=5{.}9090\ldots=5{.}\overline{90}\\\underline{-55}\\\hphantom{-}100\\\underline{-\hphantom{1}99}\\\hphantom{-10}10\\\hphantom{11}\underline{-\hphantom{1}0}\\\hphantom{-10}100\\\hphantom{11}\underline{-\hphantom{1}99}\\\hphantom{-1000}10\\\hphantom{1111}\underline{-\hphantom{1}0}\\\hphantom{-1000}100\\\hphantom{1000000}\vdots\end{array}$

$\Rightarrow\left(9\cdot0.8-0.70\right):\left(0.6+0.5\right)=5.\overline{90}$

Alternative way

Expression 1

1) Exponential Expressions

$3.1^2 = 3.1･3.1 = 9.61$

$2.2^3 = 2.2･2.2･2.2 = 10.648$

2) Multiply

$1.3･3.1^2 = 1.3･9.61 = 12.493$

$2.2^3･0.4 = 10.648･0.4 = 4.2592$

3) Add

$1.3^3･3.1+2.2^2･0.4 = 12.493+4.2592 = 16.7522$

The first expression has been evaluated to 16.7522.

Expression 2:

1) Exponential Expressions

$1.3^3 = 1.3･1.3･1.3 = 2.197$

$2.2^2 = 2.2･2.2 = 4.84$

2) Multiply

$1.3^3･3.1 = 2.197･3.1 = 6.8107$

$2.2^2･0.4 = 4.84･0.4 = 1.936$

3) Add

$1.3^3･3.1+2.2^2･0.4 = 6.8107+1.936 = 8.7467$

The second expression has been evaluated to 8.7467

Finally, the values of the two expression have to be compared.

$16.7522 > 8.7467$

The first expression is the larger one.

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:

$\frac{3}{4}=\ \frac{3\cdot25}{4\cdot25}=\frac{75}{100}$.

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:

$\frac{3}{4\ }=\frac{3\cdot25}{4\cdot25}=\frac{75}{100}=0.75$

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\cdot100}{3\cdot100}=\frac{200}{300}\ \ \$ , $\frac{400}{100}=\frac{400\cdot3}{3\cdot100}=\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 $5\cdot7=35$.

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

$\frac{2}{5}=\frac{2\cdot7}{5\cdot7}=\frac{14}{35}$, $\frac{2}{7}=\frac{2\cdot5}{7\cdot5}=\frac{10}{35}$ .

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\cdot10\ =\ 70\$.

3) Multiplying and dividing:

$\ \ \frac{1}{7}=\frac{1\cdot10}{7\cdot10}=\frac{10}{70}\$, $\frac{7}{10}=\frac{7\cdot7}{10\cdot7}=\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:

$\def\arraystretch{1.25} \begin{array}{l}\;\;\;1\;.5\;0\\ \underline{+3\;.4\;4\\}\\ \;\;\;4\;.9\;4\end{array}$

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\cdot10\cdot7\ =\ 630\$. This is the common denominator.

3) Multiply and divide.

$\frac{4}{9}\ =\ \frac{4\cdot10\cdot7}{9\cdot10\cdot7}\ =\frac{280}{630}\$,

$\frac{2}{10}=\ \frac{2}{10\cdot9\cdot7}=\frac{126}{630}\$,

$\frac{2}{7}=\frac{2\cdot10\cdot9}{7\cdot10\cdot9}\ =\ \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\cdot25}{4\cdot25}=\frac{25}{100}=0.25\$.

$\frac{3}{8}=\frac{3\cdot125}{8\cdot125}=\frac{375}{1000}=0.375\$.

2) Add, using column addition.

$\def\arraystretch{1.25} \begin{array}{l}\;\;\;0\;.2\;5\;0\\ \;\;\;3\;.2\;0\;0\\ \;\;\;0\;.3\;7\;5\\ \underline{+1_1.7_15\;0\\}\\ \;\;\;5\;.5\;2\;5\end{array}$

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\cdot10=\ 70\$.

3) Multiply and divide.

$\ \ \frac{5}{7}=\ \frac{5\cdot10}{7\cdot10}=\frac{50}{70}\$.

$\ \ \frac{3}{10}=\frac{3\cdot7}{10\cdot7}=\frac{21}{70}\$.

$\frac{2}{5}=\frac{2\cdot2\cdot7}{5\cdot2\cdot7}=\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}\$.

An adequate knowledge of exponential expressions, of the order of operations (PEMDAS) and of division by decimals is a prerequisite for working out the value of the above expression.

The following steps will guide you to the solution.

1. Work out the value of the exponential expressions.

2. Subtract.

3. Divide the difference by the decimal.

So, let's get going.

1. Work out the value of the exponential expressions.

$2.3^2=2.3\cdot2.3=5.29\$.

$\ 1.7^2=1.7\cdot1.7=2.89$ .

2. Subtract.

$2.3^2-1.7^2=5.29-2.89=2.4\$.

3. Divide the difference by 0.6.

$\ \ \ \ \left(2.3^2-1.7^2\right):\ 0.6\ =\ 2.4\ :\ 0.6\ =\ 24\ :\ 6\ =\ 4$ .

Done! $\left(\ 2.3^2-1.7^2\right):0.6\ \$evaluates to 4.

Like all other numerical expressions in exercise 4, the above contains parentheses and two decimals raised to the power of 2. Hence an adequate knowledge of the order of operations (PEMDAS), exponential expressions and division by a decimal is a prerequisite for evaluating this numerical expression.

Adhering to the following steps will guide you to the solution. This sequence of steps is the same for each numerical expression of exercise 4.

1. Evaluate the exponential expressions.

2. Subtract.

3. Divide the difference by the decimal.

So, let's get going.

1. Evaluate the exponential expressions.

$\ \ \ 5.3^2=5.3\cdot5.3=28.09\$.

$\ \ \ 1.9^2=1.9\cdot1.9=3.61\$.

2. Subtract.

$\ \ 28.09-3.61=24.48\$.

3. Divide the difference by 7.2 .

$\ \ \ 24.48\ :\ 7.2\ =\ 244.8\ :\ 72\ =\left(216+28.8\right)\ :\ 72\ =\ 3+0.4\ =\ 3.4\$.

Done! So, $\ \ \left(5.3^2-1.9^2\right)\ :\ 7.2\ =\ \left(24.48-3.61\right)\ :\ 7.2\ =24.48\ :\ 7.2=3.4\$.

Hence $\ \left(5.3^2-1.9^2\right)\ .\ 7.2\ \$evaluates to 3.4 .

$0.123+x=1\\$

Step 1: Not applicable. There are no like-terms on each side of the equality sign.

Step 2: Collect and combine like-terms of equations on opposite sides of the equality sign using the inverse operation. Variable terms are on the left-hand-side of the equality sign.

$\ \ \ 0.123+x-0.123\ =\ 1-0.123$

$\ \ \ x\ =\ 0.877$

Step 3: Not applicable. The factor of the variable term is 1.

Step 4: Check.

$0.123+0.877\ =\ 1$

$\ \ \ 1\ =\ 1$. This is a true statement.

$x-0.044=3\\$

Forme den Term nach $x$ um indem du auf beiden Seiten $0.044$ addierst.

$x=3.044$

$12.3+\left(-11.34\right)+82.7$

Using the communative property of addition, the sum can be rewritten:

$12.3+82.7+\left(-11.34\right)$

Now adding by endings and composing a multiple of 10 can be used:

$3+7=10,\ hence\ 0.3+0.7=1$

Therefore: $12.3+82.7=12+82+1=95$

So: $12.3+82.7+\left(-11.34\right)=95+\left(-11.34\right)$

$\ \ \ 11.34\ \$can be decomposed in $11\ and\ 34\ hundredths$

Using this decomposition, $\ \ 95+\left(-11.34\right)\$can be rewritten (using subtraction by place value):

$\ \ \ 95-11-0.34=84-0.34=83.66\ \$

Therefore:

$12.3+\left(-11.34\right)+82.7=83.66$

Working efficiently involves using the commutative and associative properties of addition. Therefore $\ \ \ -11.77+3.2-5.23\ \$is rewritten as a sum with two negative decimals:

$\ \ \ -11.77+3.2-5.23\ =-11.77+3.2+\left(-5.23\right)\ \$

Using the commutative property of addition $\ \ \ -11.77+3.2+\left(-5.23\right)\ \$can be rewritten:

$\ \ \ -11.77+\left(-5.23\right)+3.2\ \$

Arithmetic skills come into play as well:

Using composing a multiple of 10 together with decomposing the negative decimals and the commutative property of addition, the negative decimals can be added easily:

$\ \ \ 77+23=100,\ \ \ hence\ \ 0.77+0.23=1\ \left(77\ hundredths\ added\ to\ \ 23\ hundredths\right)\ \$

Therefore:

$-11.77+\left(-5.23\right)=-11+\left(-0.77\right)+\left(-5\right)+\left(-0.23\right)=-11+\left(-5\right)+\left(-0.77\right)+\left(-0.23\right)$

$=-16+\left(-1\right)=-17$

Therefore:

$\ \ \ -11.77+3.2-5.23\ =\ -17+3.2$

$\ \ \ -17+3.2=-17+3+0.2=-14+0.2=-13.8$

Therefore:

$\ \ \ -11.77+3.2-5.23=-13.8$

Working efficiently involves using the commutative and associative property of addition. Therefore $2.89-2.1-4.9$ is rewritten as a sum with two negative decimals:

$\ \ \ 2.89+\left(-2.1\right)+\left(-4.9\right)$

Now, the associative property of addition can be used.

$\ \ \ 2.89+\left(-2.1+\left(-4.9\right)\right)$

Next, composing a multiple of 10 together with the decomposition of the negative decimals and the commutative property of addition comes into play:

$\ \ \ 1+9=10,\ \ hence\ \ \ 0.1+0.9\ =\ 1\ \ \ \left(1\ tenths\ added\ to\ 9\ tenths\right)$

Decomposing the negative decimals and using the commutative property of addition:

$\ \ \ -2{,}1+\left(-4.9\right)=-2+\left(-0.1\right)+\left(-4\right)+\left(-0.9\right)=-2+\left(-4\right)+\left(-0.1\right)+\left(-0.9\right)=-7$

Therefore:

$\ \ \ 2.89-2.1-4.9=2.89+\left(-7\right)=2+0.89+\left(-7\right)=-5+0.89$

Using composing a multiple of 10 again

$89+11=100,\ \ \ hence\ \ \ 0.89+0.11\ =\ 1\ \ \left(89\ hundredths\ added\ to\ 11\ hundredths\right)$

$-5+0.89=-4+\left(-1\right)+0.89=-4+\left(-0.11\right)=-4.11\$

Therefore:

$\ \ \ 2.89-2.1-4.9=-4.11$