Understanding Fractions and Completing the Whole

The question asks us to find a fraction which, when added to $\frac{5}{8}$, results in the sum being equal to 1. The number 1 represents a whole.

Let the unknown fraction be represented by $x$. We can write the problem as an equation:

$\frac{5}{8} + x = 1$

To find the value of $x$, we need to isolate it. We can do this by subtracting $\frac{5}{8}$ from both sides of the equation:

$x = 1 - \frac{5}{8}$

To subtract a fraction from a whole number (1), we need to express the whole number 1 as a fraction with the same denominator as the fraction being subtracted. In this case, the denominator is 8. So, 1 can be written as $\frac{8}{8}$.

Now, the equation becomes:

$x = \frac{8}{8} - \frac{5}{8}$

Subtracting fractions with the same denominator is straightforward. We subtract the numerators and keep the denominator the same:

$x = \frac{8 - 5}{8}$

$x = \frac{3}{8}$

So, the fraction that needs to be added to $\frac{5}{8}$ to get 1 is $\frac{3}{8}$.

Now, let's look at the given options and see which one is equivalent to $\frac{3}{8}$.

• Option 1: $\frac{6}{24}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 6. $\frac{6 \div 6}{24 \div 6} = \frac{1}{4}$. Is $\frac{1}{4}$ equal to $\frac{3}{8}$? No, because $\frac{1}{4}$ is equivalent to $\frac{2}{8}$ (by multiplying numerator and denominator by 2).
• Option 2: $\frac{5}{2}$. This is an improper fraction, meaning its value is greater than 1 ($5 \div 2 = 2.5$). Adding a fraction greater than 1 to $\frac{5}{8}$ will result in a number much greater than 1. So, this is not the correct fraction.
• Option 3: $\frac{6}{16}$. We can simplify this fraction by dividing both the numerator and the denominator by their greatest common divisor, which is 2. $\frac{6 \div 2}{16 \div 2} = \frac{3}{8}$. This fraction is exactly equal to the fraction we calculated ($\frac{3}{8}$).
• Option 4: $\frac{6}{3}$. This fraction simplifies to a whole number. $\frac{6}{3} = 2$. Adding 2 to $\frac{5}{8}$ will result in a number much greater than 1. So, this is not the correct fraction.

Comparing the options to our calculated fraction $\frac{3}{8}$, we see that Option 3, $\frac{6}{16}$, is equivalent to $\frac{3}{8}$.

Let's check our answer:

$\frac{5}{8} + \frac{6}{16} = \frac{5}{8} + \frac{3}{8} = \frac{5+3}{8} = \frac{8}{8} = 1$

The sum is indeed 1, confirming that $\frac{6}{16}$ is the correct fraction.

The final answer is $\frac{6}{16}$.

Revision Table: Key Concepts for Fraction Addition

Understanding how to add and subtract fractions is crucial. Here’s a quick review:

Additional Information: Fractions and the Number 1

The number 1 plays a special role in fractions. Any fraction where the numerator is equal to the denominator (and the denominator is not zero) is equivalent to 1. Examples include $\frac{2}{2}$, $\frac{5}{5}$, $\frac{8}{8}$, $\frac{100}{100}$, etc.

When we talk about "completing" a fraction to make 1, we are essentially finding the difference between 1 and the given fraction. This is like finding the missing piece of a pie to make a whole pie.

For example, if you have $\frac{5}{8}$ of a pie, you need $1 - \frac{5}{8} = \frac{8}{8} - \frac{5}{8} = \frac{3}{8}$ more pie to have a whole pie. This illustrates the concept used in solving the problem.

Understanding equivalent fractions is also vital. Fractions like $\frac{3}{8}$, $\frac{6}{16}$, $\frac{9}{24}$, etc., all represent the same value, even though they have different numerators and denominators. This is because the ratio between the numerator and denominator is the same.