SSC GD Decimals & Fractions

• Types of Fractions
  • Proper, Improper, and Mixed Fractions
  • Equivalent Fractions
• Turning Fractions into Decimals
• Turning Decimals back into Fractions
• Adding and Subtracting Fractions (The LCM Rule)
• Multiplying and Dividing Fractions
• Working with Decimals (Addition, Subtraction, Multiplication, Division)

Types of Fractions

Proper, Improper, and Mixed Fractions

Welcome! We are starting our study notes for SSC GD Decimals & Fractions. A fraction is just a part of a whole thing. Think of cutting a pizza.

• A fraction has two parts: the top number (Numerator) and the bottom number (Denominator).
• The denominator tells you how many total pieces the whole thing was cut into.
• The numerator tells you how many pieces you have.

1. Proper Fractions

• The top number (Numerator) is smaller than the bottom number (Denominator).
• Example: $\frac{1}{2}$ or $\frac{3}{5}$.
• Why: This fraction is always less than 1 whole. You have fewer pieces than the total needed for a whole pizza.

2. Improper Fractions

• The top number (Numerator) is equal to or larger than the bottom number (Denominator).
• Example: $\frac{5}{4}$ or $\frac{7}{3}$.
• Why: This fraction is 1 whole or more than 1 whole. You have more pieces than one pizza can hold, so you must have opened a second pizza box!

3. Mixed Fractions

• This is a whole number and a proper fraction put together.
• Example: $1\frac{1}{4}$ (This means 1 whole thing and $\frac{1}{4}$ of another thing).
• We use mixed fractions to write improper fractions simply. This is a key skill for solving SSC GD Decimals & Fractions Question pdf problems.

Example 1: Convert the improper fraction $\frac{11}{3}$ into a mixed fraction.

Solution:

1. We divide the numerator (11) by the denominator (3).
2. $11÷3=3$ with a remainder of 2.
3. The whole number is 3, the new numerator is the remainder (2), and the denominator stays the same (3). So, the answer is $3\frac{2}{3}$.

Example 2: Convert the mixed fraction $4\frac{1}{5}$ into an improper fraction.

Solution:

1. We multiply the whole number (4) by the denominator (5): $4\times 5=20$.
2. We add the numerator (1) to that result: $20+1=21$.
3. The new numerator is 21, and the denominator stays the same (5). So, the answer is $\frac{21}{5}$.

Example 3: Identify the type of fraction: $\frac{15}{16}$.

Solution:

1. We compare the numerator (15) and the denominator (16).
2. Since $15<16$, the numerator is smaller.
3. This is a Proper Fraction.

Example 4: Identify the type of fraction: $2\frac{5}{8}$.

Solution:

1. The fraction has a whole number (2) and a fraction ($\frac{5}{8}$).
2. Any fraction written with a whole number part is a Mixed Fraction.

Equivalent Fractions

• Equivalent means “equal in value.”
• Equivalent fractions look different but represent the exact same amount.
• Example: $\frac{1}{2}$ is the same as $\frac{2}{4}$ and $\frac{50}{100}$.
• Why: If you multiply or divide the numerator and the denominator by the same non-zero number, the value does not change.

Turning Fractions into Decimals

Decimals are just another way to write fractions, especially those with denominators of 10, 100, 1000, and so on. Mastering this conversion is vital for SSC GD Decimals & Fractions.

Method 1: Making the Denominator a Power of 10

• If you can easily change the denominator to 10, 100, or 1000, this is the fastest way.
• Example: $\frac{3}{5}$. We can multiply 5 by 2 to get 10.
• We must multiply the numerator by 2 as well: $\frac{3\times 2}{5\times 2}=\frac{6}{10}$.
• $\frac{6}{10}$ means 6 tenths, which is written as $0.6$.

Method 2: Division

• If the denominator cannot easily become 10 or 100, you must divide the numerator by the denominator.
• Example: $\frac{1}{8}$. We divide 1 by 8.

Example 1: Convert $\frac{7}{20}$ to a decimal using the power of 10 method.

Solution:

1. We want the denominator (20) to become 100. We multiply $20\times 5=100$.
2. We must multiply the numerator (7) by 5: $7\times 5=35$.
3. The new fraction is $\frac{35}{100}$.
4. $\frac{35}{100}$ means the decimal point moves two places to the left. The answer is $0.35$.

Example 2: Convert $\frac{3}{4}$ to a decimal.

Solution:

1. We can make the denominator 100: $4\times 25=100$.
2. Multiply the numerator by 25: $3\times 25=75$.
3. The fraction is $\frac{75}{100}$.
4. The decimal form is $0.75$.

Example 3: Convert $\frac{5}{8}$ to a decimal using division.

Solution:

1. We divide 5 by 8.
2. $5÷8$. Since 8 doesn’t go into 5, we add a decimal point and a zero (5.0).
3. $50÷8=6$ (Remainder 2). We write down 0.6.
4. Bring down another zero (20). $20÷8=2$ (Remainder 4). We write down 0.62.
5. Bring down another zero (40). $40÷8=5$ (Remainder 0).
6. The answer is $0.625$.

Example 4: What is the decimal equivalent of $1\frac{1}{2}$?

Solution:

1. We know the whole number is 1, so the decimal starts with 1.
2. We convert the fraction $\frac{1}{2}$. $1÷2=0.5$.
3. We combine the whole number and the decimal part: $1+0.5=1.5$.
4. The answer is $1.5$.

Turning Decimals back into Fractions

This is the reverse process and is very important for simplifying answers in SSC GD Decimals & Fractions Question pdf.

The Rule:

1. Write the decimal number without the decimal point as the numerator.
2. For the denominator, write 1 followed by as many zeros as there are digits after the decimal point.
3. Always simplify the resulting fraction to its lowest terms.

Example 1: Convert $0.4$ to a fraction and simplify.

Solution:

1. The numerator is 4 (the number without the point).
2. There is one digit after the decimal point (4), so the denominator is 10. The fraction is $\frac{4}{10}$.
3. We simplify by dividing both by 2: $\frac{4÷2}{10÷2}=\frac{2}{5}$.
4. The simplified answer is $\frac{2}{5}$.

Example 2: Convert $1.25$ to a mixed fraction.

Solution:

1. The whole number is 1. We focus on the decimal part: $0.25$.
2. The numerator is 25. There are two digits after the point, so the denominator is 100. The fraction is $\frac{25}{100}$.
3. Simplify $\frac{25}{100}$. We divide both by 25: $\frac{25÷25}{100÷25}=\frac{1}{4}$.
4. We combine the whole number and the fraction. The answer is $1\frac{1}{4}$.

Example 3: Convert $0.008$ to a fraction and simplify.

Solution:

1. The numerator is 8 (ignoring the leading zeros).
2. There are three digits after the point (0, 0, 8), so the denominator is 1000. The fraction is $\frac{8}{1000}$.
3. Simplify by dividing both by 8: $\frac{8÷8}{1000÷8}=\frac{1}{125}$.
4. The simplified answer is $\frac{1}{125}$.

Example 4: Convert $3.6$ to an improper fraction.

Solution:

1. We write the entire number without the decimal point as the numerator: 36.
2. There is one digit after the point, so the denominator is 10. The fraction is $\frac{36}{10}$.
3. Simplify by dividing both by 2: $\frac{36÷2}{10÷2}=\frac{18}{5}$.
4. The improper fraction answer is $\frac{18}{5}$.

Adding and Subtracting Fractions (The LCM Rule)

You cannot add or subtract fractions unless they have the same denominator. This is the most common mistake in SSC GD Decimals & Fractions problems.

• Why? Imagine you have $\frac{1}{2}$ of a pizza and $\frac{1}{4}$ of a cake. You cannot just add $1+1=2$ pieces because the sizes of the pieces are different!
• We use the Least Common Multiple (LCM) to find a common denominator.

Steps for Addition/Subtraction:

1. Find the LCM of the denominators.
2. Change both fractions to equivalent fractions using the LCM as the new denominator.
3. Add or subtract the numerators.
4. Keep the common denominator.
5. Simplify the final answer.

Example 1: Add the fractions: $\frac{1}{3}+\frac{1}{6}$.

Solution:

1. Find the LCM of 3 and 6. The LCM is 6.
2. Change $\frac{1}{3}$ to have a denominator of 6: $\frac{1\times 2}{3\times 2}=\frac{2}{6}$.
3. Now add: $\frac{2}{6}+\frac{1}{6}$.
4. Add the numerators: $2+1=3$. Keep the denominator 6. The result is $\frac{3}{6}$.
5. Simplify: $\frac{3}{6}=\frac{1}{2}$.

Example 2: Subtract the fractions: $\frac{5}{8}-\frac{1}{4}$.

Solution:

1. Find the LCM of 8 and 4. The LCM is 8.
2. Change $\frac{1}{4}$ to have a denominator of 8: $\frac{1\times 2}{4\times 2}=\frac{2}{8}$.
3. Now subtract: $\frac{5}{8}-\frac{2}{8}$.
4. Subtract the numerators: $5-2=3$. Keep the denominator 8.
5. The answer is $\frac{3}{8}$.

Example 3: Calculate: $2\frac{1}{2}+3\frac{1}{3}$.

Solution:

1. Convert mixed fractions to improper fractions: $2\frac{1}{2}=\frac{5}{2}$ and $3\frac{1}{3}=\frac{10}{3}$.
2. Find the LCM of 2 and 3. The LCM is 6.
3. Convert fractions: $\frac{5}{2}=\frac{5\times 3}{2\times 3}=\frac{15}{6}$ and $\frac{10}{3}=\frac{10\times 2}{3\times 2}=\frac{20}{6}$.
4. Add the new fractions: $\frac{15}{6}+\frac{20}{6}=\frac{35}{6}$.
5. Convert back to mixed fraction: $35÷6=5$ remainder 5. The answer is $5\frac{5}{6}$.

Example 4: Find the difference: $\frac{7}{10}-\frac{1}{5}$.

Solution:

1. Find the LCM of 10 and 5. The LCM is 10.
2. Change $\frac{1}{5}$ to have a denominator of 10: $\frac{1\times 2}{5\times 2}=\frac{2}{10}$.
3. Subtract: $\frac{7}{10}-\frac{2}{10}$.
4. Subtract the numerators: $7-2=5$. The result is $\frac{5}{10}$.
5. Simplify: $\frac{5}{10}=\frac{1}{2}$.

Multiplying and Dividing Fractions

These operations are often easier than addition or subtraction because you do not need a common denominator. This is a common area tested in the SSC GD Decimals & Fractions Question pdf.

Multiplication

• Multiply the numerators together.
• Multiply the denominators together.
• Simplify the result.

Example 1: Multiply: $\frac{2}{3}\times \frac{5}{7}$.

Solution:

1. Multiply the numerators: $2\times 5=10$.
2. Multiply the denominators: $3\times 7=21$.
3. The result is $\frac{10}{21}$. (Cannot be simplified further).

Example 2: Calculate: $\frac{3}{4}\times 8$.

Solution:

1. Write the whole number 8 as a fraction: $\frac{8}{1}$. The problem is $\frac{3}{4}\times \frac{8}{1}$.
2. Multiply numerators: $3\times 8=24$.
3. Multiply denominators: $4\times 1=4$. The result is $\frac{24}{4}$.
4. Simplify: $24÷4=6$. The answer is 6.

Division

• Keep, Change, Flip (KCF):
  1. Keep the first fraction as it is.
  2. Change the division sign ($÷$) to a multiplication sign ($\times$).
  3. Flip the second fraction (find its reciprocal).
• Then, multiply as usual.

Example 3: Divide: $\frac{1}{5}÷\frac{2}{3}$.

Solution:

1. Keep the first fraction: $\frac{1}{5}$.
2. Change the sign: $\times$.
3. Flip the second fraction ($\frac{2}{3}$ becomes $\frac{3}{2}$). The problem is now $\frac{1}{5}\times \frac{3}{2}$.
4. Multiply numerators: $1\times 3=3$.
5. Multiply denominators: $5\times 2=10$. The answer is $\frac{3}{10}$.

Example 4: Calculate: $2\frac{1}{4}÷\frac{3}{8}$.

Solution:

1. Convert the mixed fraction to improper: $2\frac{1}{4}=\frac{9}{4}$. The problem is $\frac{9}{4}÷\frac{3}{8}$.
2. Keep, Change, Flip: $\frac{9}{4}\times \frac{8}{3}$.
3. Multiply: $\frac{9\times 8}{4\times 3}=\frac{72}{12}$.
4. Simplify: $72÷12=6$. The answer is 6.

Working with Decimals (Addition, Subtraction, Multiplication, Division)

Decimals are used frequently in the quantitative section of the SSC GD exam. We must handle the decimal point correctly.

Addition and Subtraction

• The Golden Rule: Always line up the decimal points perfectly.
• You can add zeros to the end of a decimal number without changing its value (e.g., $1.5=1.500$). This helps keep the columns straight.

Example 1: Add: $12.3+5.07+0.9$.

Solution:

1. Line up the decimal points and add trailing zeros so all numbers have two decimal places: $12.30$ $5.07$ $+0.90$
2. Add the columns starting from the right: $0+7+0=7$.
3. Next column: $3+0+9=12$. Write down 2, carry 1.
4. Place the decimal point.
5. Next column: $1(\text{carry})+2+5+0=8$.
6. Last column: $1$.
7. The answer is $18.27$.

Example 2: Subtract: $25.1-13.45$.

Solution:

1. Line up the decimal points and add a zero to 25.1: $25.10$ $-13.45$
2. Subtract the rightmost column (borrowing): $10-5=5$.
3. Next column (borrowing again): $10-4=6$.
4. Place the decimal point.
5. Next column: $4-3=1$.
6. Last column: $2-1=1$.
7. The answer is $11.65$.

Multiplication

• Ignore the decimal point during the multiplication process.
• Count the total number of digits after the decimal point in all the numbers being multiplied.
• In the final answer, count that many places from the right and place the decimal point.

Example 3: Multiply: $1.2\times 0.03$.

Solution:

1. Multiply the numbers without the decimal point: $12\times 3=36$.
2. Count the decimal places: $1.2$ has one place. $0.03$ has two places. Total places $=1+2=3$.
3. Start from the right of 36 and move the point 3 places to the left. We need to add a zero: $0.036$.
4. The answer is $0.036$. This is a common type of SSC GD Decimals & Fractions problem.

Division

• The Goal: You cannot divide by a decimal. You must make the divisor (the number outside the division box) a whole number.
• Move the decimal point in the divisor to the right until it is a whole number.
• Move the decimal point in the dividend (the number inside the division box) the exact same number of places to the right.
• Place the decimal point in the answer directly above its new position in the dividend.

Example 4: Divide: $4.8÷0.06$.

Solution:

1. The divisor is $0.06$. We must move the point 2 places right to make it 6.
2. We move the point in the dividend (4.8) 2 places right. $4.8$ becomes $480$.
3. The new problem is $480÷6$.
4. $480÷6=80$.
5. The answer is 80. This completes our SSC GD Decimals & Fractions study notes.