Informative line

# Decimals and their Place Values

• Decimal is a part of whole.
• Decimal numbers have wholes, a decimal point and parts in it.

Example: $$1.50$$

Each digit in a decimal number has a place value.

Example$$15.75$$

 Hundreds Tens Ones Decimal Point Tenths Hundredths Thousandths Ten Thousandths Hundred Thousandths Millionths 1 5 . 7 5

• We can also write decimal numbers in words, by naming the place value of the last digit.

Examples:

1. Seventy-two and one-tenths $$=72.1$$
2. One and two hundred fifteen thousandths $$=1.215$$
3. $$25.002=$$ Twenty-five and two-thousandths
4. $$2.5=$$ Two and five-tenths

#### Which one of the following options represents forty-nine and fifty-eight thousandths, in numbers?

A $$49.58$$

B $$49.058$$

C $$58.49$$

D $$58.049$$

×

In forty-nine and fifty-eight thousandths, forty-nine is the whole and fifty-eight thousandths is the parts.

Forty-nine $$=49$$

Fifty-eight thousandths $$=0.058$$

$$\therefore$$ The number is $$49.058$$.

Hence, option (B) is correct.

### Which one of the following options represents forty-nine and fifty-eight thousandths, in numbers?

A

$$49.58$$

.

B

$$49.058$$

C

$$58.49$$

D

$$58.049$$

Option B is Correct

# Subtraction of Decimals

• Subtraction of decimals can be easily done by dealing with the wholes and the parts of the numbers separately.

Example: $$5.674-2.5$$

• Here, both decimals have different place values, so we can add zeros to make them same.

$$5.674-2.500$$

• Now, both decimal numbers have same place values.

Next, we write the decimal numbers vertically, lining up the decimal points and each digit according to its place value.

$$\begin{array}\\ &5&.&6&7&4\\ -&2&.&5&0&0\\ \hline\\ \hline \end{array}$$

• Now, subtracting the columns vertically.

$$\begin{array}\\ &5&.&6&7&4\\ -&2&.&5&0&0 \\ \hline &3&.&1&7&4 \\\hline \end{array}$$

• Difference of $$5.674$$ and $$2.5$$ is $$3.174$$

#### Find:​ $$10.1-2.85$$

A $$8.15$$

B $$12.95$$

C $$7.25$$

D $$12.86$$

×

Since both numbers have different place values, so we add zero to make them same.

$$10.10-2.85$$

Now, we write the decimals vertically, lining up the decimal points and each digit according to its place value.

$$\begin{array}\\ &1&0&.&1&0\\ -&&2&.&8&5\\ \hline\\ \hline \end{array}$$

On subtracting the columns vertically, we get

$$\begin{array}\\ &1&0&.&1&0\\ -&&2&.&8&5\\ \hline&&7&.&2&5\\ \hline \end{array}$$

Hence, option (C) is correct.

### Find:​ $$10.1-2.85$$

A

$$8.15$$

.

B

$$12.95$$

C

$$7.25$$

D

$$12.86$$

Option C is Correct

# Multiplication of Decimals

• Factors are referred to as the numbers which are being multiplied.
• Product refers to the result of a multiplication problem.
• To multiply two decimal numbers, first line them up in right alignment.
• Then, multiply each digit of multiplier by each digit of multiplicand.
• Count the number of digits after the decimal point in multiplier and multiplicand.
• The product will have same number, of total digits, after the decimal point.

Example: $$5.23×2.1$$

• First, line up the decimals in right alignment.

$$\begin{array}\\ &5&.&2&3\\ ×&&2&.&1\\ \hline\\ \hline \end{array}$$

• Now, multiply each digit of the multiplier by each digit of the multiplicand, just like whole numbers, by ignoring the decimal point.

$$\begin{array}\\ &&5&.&2&3\ \rightarrow2 \text{ digits}\\ ×&&&2&.&1\ \rightarrow1 \text{ digit}\\ \hline &&&5&2&3\\ +&1&0&4&6&0&\\ \hline &1&0&9&8&3&\\ \end{array}$$

• There are $$2$$ digits after the decimal point in multiplicand and $$1$$ digit in multiplier.
• So, the product will have $$3$$ digits after the decimal point.

$$10.983$$

• Thus, $$10.983$$ is the product of $$5.23$$ and $$2.1$$

Note:

When we multiply decimals by power of $$10$$, then we can simply move the decimal point to the right, by the number of places per multiple of $$10$$.

Example: $$6.45\times10=64.5$$

Here, we move decimal by one place value.

$$7.234×1000=7234$$

Here, we move decimal by three place values.

#### Solve: $$6.35×5.9$$

A $$37.465$$

B $$35.465$$

C $$39.465$$

D $$38.469$$

×

First, we will line up the decimal numbers in right alignment.

$$\begin{array}\\ &6&.&3&5\\ ×&&5&.&9\\ \hline\\ \hline \end{array}$$

Now, we multiply each digit of the multiplier by each digit of the multiplicand, just like whole numbers, by ignoring the decimal point.

$$\begin{array}\\ &&6&.&3&5\rightarrow2 \text{ digits}\\ ×&&&5&.&9\rightarrow1 \text{ digit}\\\hline\ &&5&7&1&5\\ +&3&1&7&5&0\\ \hline &3&7&4&6&5\\ \end{array}$$

• There are $$2$$ digits after the decimal point in the multiplicand and $$1$$ digit in the multiplier.
• So, the product will have $$3$$ digits after the decimal point.

$$37.465$$

Hence, option (A) is correct.

### Solve: $$6.35×5.9$$

A

$$37.465$$

.

B

$$35.465$$

C

$$39.465$$

D

$$38.469$$

Option A is Correct

# Representation of Decimals on Number Line

• Decimals can be represented on the number line.
• To find the decimal number on the number line, we move in the forward direction from zero.
• First, find the wholes and then the parts on the number line.

For example: We want to represent $$3.7$$ on the number line.

• To find the decimal number $$3.7$$ on the number line, find the whole $$3$$.
• To find $$3$$ on the number line, move in the forward direction and mark the whole at $$3$$.

Now, we have to find the part $$0.7$$ on the number line. To find $$0.7$$ on the number line, divide the interval between $$3$$ and $$4$$ into $$10$$ equal sections, each having a scale of $$0.1$$.

• Now, start from $$3$$ and count $$7$$ slashes by moving forward. Mark the $$7$$th slash.

• We have reached the whole $$3$$ and the parts $$0.7$$ on the number line.
• So, by combining these, we get the decimal number $$3.7$$ on the number line.

#### Which one of the following points is closest to ​2.65 on the given number line?

A Point A

B Point B

C Point C

D Point D

×

First, we need to find intervals between whole numbers.

There are four equal intervals between whole numbers, this means that each interval has a scale of $$.25$$.

So, the given number $$2.65$$ should be placed between $$2.50$$ and $$2.75$$.

Among $$2.50$$ and $$2.75,\;2.65$$ is closer to $$2.75$$.

Thus, point $$D$$ is closest to $$2.65$$.

Hence, option (D) is correct.

### Which one of the following points is closest to ​2.65 on the given number line?

A

Point A

.

B

Point B

C

Point C

D

Point D

Option D is Correct

• Addition of decimals can be easily done by dealing with the wholes and the parts of the numbers separately.

Example: Add $$3.4$$ and $$7.347$$

• Both decimal numbers have different place values, so we can add zeros to make their place value same.

$$3.400+7.347$$

• Now, both the numbers have same place values.

• Write the decimal numbers vertically, lining up the decimal points and each digit according to its place value

$$\begin{array}\\ &3&.&4&0&0\\ +&7&.&3&4&7\\ \hline\\ \hline \end{array}$$

$$\begin{array}\\ &3&.&4&0&0\\ +&7&.&3&4&7\\ \hline 1&0&.&7&4&7\\ \hline \end{array}$$

• Sum of $$3.4$$ and $$7.347$$ is $$10.747$$

#### Find:​ $$20.6+1.225+2.75$$

A $$23.575$$

B $$24.575$$

C $$24.306$$

D $$23.306$$

×

All decimal numbers have different place values, so we can add zeros to make them same.

$$20.600+1.225+2.750$$

Now, write the numbers vertically, lining up the decimal points and each digit according to its place value.

$$\begin{array}\\ &2&0&.&6&0&0\\ +&&1&.&2&2&5\\ &&2&.&7&5&0\\ \hline \\ \hline \end{array}$$

On adding the columns vertically, we get

$$\begin{array}\\ &2&0&.&6&0&0\\ +&&1&.&2&2&5\\ &&2&.&7&5&0\\ \hline &2&4&.&5&7&5\\ \hline \end{array}$$

Hence, option (B) is correct.

### Find:​ $$20.6+1.225+2.75$$

A

$$23.575$$

.

B

$$24.575$$

C

$$24.306$$

D

$$23.306$$

Option B is Correct

# Division of Decimals

• Dividing decimals by other decimals is almost same as dividing whole numbers.
• To make divisor into a whole number, multiply the divisor by a power of ten.
• If we multiply the divisor by a power of ten, then multiply the dividend also by the same power of ten.

Example: Evaluate $$7.2\div3.6$$

• First, make the divisor into a whole number.
• Multiply $$3.6$$ by $$10$$.

$$3.6×10=36$$

• Now, we also need to multiply dividend by $$10$$.

$$7.2×10=72$$

• Now, divide as whole numbers.
• So, quotient of $$7.2$$ by $$3.6$$ is $$2$$.

Note:

While dividing a decimal number by another decimal number, if a part of the quotient is repeating, then it is called repeating or recurring decimal.

Example:

We divide $$0.1$$ by $$0.3$$.

• We get $$3$$, which is repeating again and again, it is known as repeating or recurring decimal.
• It is written as $$0.\overline{3}$$
• Thus, $$0.1\div0.3=0.\overline{3}$$

Note:

When we divide decimals by power of $$10$$, then we can simply move the decimal point to the left, by the number of places per multiple of $$10$$.

Example 1: $$17.25\div1000$$

Here, we will move decimal by three place values.

Example 2: $$57.6\div10$$

Here, we will move decimal point by one place value.

#### Evaluate:

A $$1.35$$

B $$0.14$$

C $$1.14$$

D $$14$$

×

First, we make the divisor a whole number by multiplying $$.09$$ by $$100$$.

$$.09×100=9$$

Now, we also need to multiply dividend by $$100$$.

$$1.26×100=126$$

Now, we divide them as whole numbers.

Thus, $$1.26\div0.09=14$$

Hence, option (D) is correct.

### Evaluate:

A

$$1.35$$

.

B

$$0.14$$

C

$$1.14$$

D

$$14$$

Option D is Correct

# Ordering Decimals

• We can arrange decimals from least to greatest or greatest to least.
• To do so, first line up the decimal point and each digit in the place value chart.
• Then add zeros to make the same number of digits in the decimal numbers.
• Now, compare from left to right.

Example: Compare $$13.2,\;13.299,\;13.213,\;13.226$$

• Lining up each decimal point and each digit in the place value chart, we get
 Tens Ones . Tenths Hundredths Thousandths 1 3 . 2 0 0 1 3 . 2 9 9 1 3 . 2 1 3 1 3 . 2 2 6
• Now we start comparing from tens column (left most column).
• Tens, ones and tenths columns have same digits, i.e. $$1,\,3,\,2$$, respectively.
• So, we compare hundredths column.
• In the hundredths column,

$$0<1<2<9$$

• Hence, $$13.200<13.213<13.226<13.299$$

#### Arrange $$6.531,\;6.446,\;6.921,\;6.189$$ from least to greatest.

A $$6.921,\;6.531,\;6.466,\;6.189$$

B $$6.921,\;6.189,\;6.446,\;6.531$$

C $$6.189,\;6.921,\;6.446,\;6.531$$

D $$6.189,\;6.446,\;6.531,\;6.921$$

×

Line up the decimal point and each digit in the place value chart.

 Ones . Tenths Hundredths Thousandths 6 . 5 3 1 6 . 4 4 6 6 . 9 2 1 6 . 1 8 9

Start comparing from ones column.

Ones column has same number i.e. $$6$$.

So, we compare tenths column.

In the tenths column,

$$1<4<5<9$$

Thus, $$6.189,\;6.446,\;6.531,\;6.921$$

Hence, option (D) is correct.

### Arrange $$6.531,\;6.446,\;6.921,\;6.189$$ from least to greatest.

A

$$6.921,\;6.531,\;6.466,\;6.189$$

.

B

$$6.921,\;6.189,\;6.446,\;6.531$$

C

$$6.189,\;6.921,\;6.446,\;6.531$$

D

$$6.189,\;6.446,\;6.531,\;6.921$$

Option D is Correct

# Word Problems (Two or More Operations)

• Decimal numbers are used in real world problems too.
• There are countless examples which show that 'How to deal with decimals in real life?'
• Consider an example to understand real world problems.
• Alex needs $$6$$ paint brushes and some colors for drawing. He has $$12.5$$ to spend and takes $$10$$ more from his father. If the cost of one paint brush is $$1.2$$, then how much money he is left with to buy colors?
• We can solve these type of problems by going step by step, using suitable operations.
• Cost of one paint brush is $$1.2$$ and there are $$6$$ brushes.

$$\therefore$$ We need to multiply $$1.2$$ by $$6$$ to find out the cost of $$6$$ brushes.

$$=1.2×6$$

$$=7.2$$

So, cost of $$6$$ brushes $$=7.2$$

Alex has $$12.5$$ and he takes $$10$$ more from his father.

$$\therefore$$ Total money Alex has $$=12.5+10$$

$$=22.5$$

• Since he spent $$7.2$$ for brushes, so we need to subtract $$7.2$$ from the total money, i.e. $$22.5$$

$$=22.5-7.2$$

$$=15.3$$

Thus, he is left with $$15.3$$ to buy colors.

#### Ms. Wendy has $$0.8$$ liters of energy drink, she adds twice of the quantity of energy drink that she already had. If she distributes total quantity of energy drink to her $$9$$ players, then find the quantity of energy drink that each player gets?

A $$\text{0.26 liters}$$

B $$\text{0.8 liters}$$

C $$\text{2.4 liters}$$

D $$\text{1.8 liters}$$

×

Given:

Energy drink that Ms. Wendy has $$=0.8$$ liters

Twice of the energy drink that she had

$$=2×0.8$$

$$=1.6$$ liters

Total energy drink Ms. Wendy has, after adding twice of the energy drink that she had

$$=0.8+1.6$$

$$=2.4$$ liters

Now according to the situation, she distributes energy drink among her $$9$$ players.

$$=2.4\div9$$

$$=0.26$$ liters

Thus, each player gets $$0.26$$ liters of energy drink.

Hence, option (A) is correct.

### Ms. Wendy has $$0.8$$ liters of energy drink, she adds twice of the quantity of energy drink that she already had. If she distributes total quantity of energy drink to her $$9$$ players, then find the quantity of energy drink that each player gets?

A

$$\text{0.26 liters}$$

.

B

$$\text{0.8 liters}$$

C

$$\text{2.4 liters}$$

D

$$\text{1.8 liters}$$

Option A is Correct