Understanding the Fraction Problem

The question asks us to find out what number needs to be added to the fraction $\frac{4}{5}$ to get the fraction $\frac{5}{4}$. Let the number we need to add be represented by $x$. We can write this relationship as an equation:

$\frac{4}{5} + x = \frac{5}{4}$

To find the value of $x$, we need to isolate it. We can do this by subtracting $\frac{4}{5}$ from both sides of the equation:

$x = \frac{5}{4} - \frac{4}{5}$

Solving the Fraction Subtraction

To subtract fractions with different denominators, we need to find a common denominator. The denominators are 4 and 5. The least common multiple (LCM) of 4 and 5 is 20. We will convert both fractions to equivalent fractions with a denominator of 20.

Converting $\frac{5}{4}$

To change the denominator of $\frac{5}{4}$ to 20, we multiply the denominator (4) by 5 to get 20. We must also multiply the numerator (5) by the same number (5) to keep the fraction equivalent:

$\frac{5}{4} = \frac{5 \times 5}{4 \times 5} = \frac{25}{20}$

Converting $\frac{4}{5}$

To change the denominator of $\frac{4}{5}$ to 20, we multiply the denominator (5) by 4 to get 20. We must also multiply the numerator (4) by the same number (4) to keep the fraction equivalent:

$\frac{4}{5} = \frac{4 \times 4}{5 \times 4} = \frac{16}{20}$

Subtracting the Equivalent Fractions

Now that both fractions have the same denominator (20), we can subtract them:

$x = \frac{25}{20} - \frac{16}{20}$

To subtract fractions with the same denominator, we subtract the numerators and keep the denominator the same:

$x = \frac{25 - 16}{20} = \frac{9}{20}$

So, the number that needs to be added to $\frac{4}{5}$ to obtain $\frac{5}{4}$ is $\frac{9}{20}$.

Checking the Options

Let's compare our result with the given options:

• Option 1: 16/20
• Option 2: 1.25/0.8 (This is a division of decimals, not a simple fraction answer in the required format)
• Option 3: -1
• Option 4: 9/20

Our calculated value, $\frac{9}{20}$, matches Option 4.

Summary of Steps

1. Set up the equation based on the problem statement: $\frac{4}{5} + x = \frac{5}{4}$.
2. Rearrange the equation to solve for $x$: $x = \frac{5}{4} - \frac{4}{5}$.
3. Find a common denominator for the fractions $\frac{5}{4}$ and $\frac{4}{5}$, which is 20.
4. Convert $\frac{5}{4}$ to an equivalent fraction with denominator 20: $\frac{25}{20}$.
5. Convert $\frac{4}{5}$ to an equivalent fraction with denominator 20: $\frac{16}{20}$.
6. Subtract the equivalent fractions: $\frac{25}{20} - \frac{16}{20} = \frac{9}{20}$.
7. Identify the correct option.

Revision Table: Fraction Addition and Subtraction

Additional Information: Working with Fractions

Fractions are a fundamental part of mathematics. They represent parts of a whole or a ratio between two numbers. The top number is the numerator, and the bottom number is the denominator.

• The denominator tells us how many equal parts the whole is divided into.
• The numerator tells us how many of those parts we have.
• Adding or subtracting fractions requires them to refer to the same size parts, which is why finding a common denominator is crucial.
• Multiplying fractions is simpler: multiply numerators and multiply denominators.
• Dividing fractions involves flipping the second fraction (finding its reciprocal) and then multiplying.

Understanding these basic operations is key to solving more complex problems involving fractions.