Write $\frac{4 y}{5}+\frac{y}{10}$ as a single fraction in its simplest form.
Answer (2)
Find the least common denominator (LCD) of 5 and 10, which is 10.
Rewrite 5 4 y with the LCD: 5 4 y = 10 8 y .
Add the fractions: 10 8 y + 10 y = 10 9 y .
The simplified fraction is 10 9 y .
Explanation
Understanding the Problem We are asked to write the expression 5 4 y + 10 y as a single fraction in its simplest form. This involves finding a common denominator and adding the fractions.
Finding a Common Denominator To add the two fractions, we need to find a common denominator. The least common denominator (LCD) of 5 and 10 is 10. We rewrite the first fraction with the LCD of 10: 5 4 y = 5 × 2 4 y × 2 = 10 8 y . The second fraction already has a denominator of 10, so it remains as 10 y .
Adding the Fractions Now we can add the two fractions: 10 8 y + 10 y = 10 8 y + y = 10 9 y .
Simplifying the Fraction Finally, we check if the resulting fraction can be simplified further. Since 9 and 10 have no common factors other than 1, the fraction 10 9 y is already in its simplest form.
Final Answer Therefore, the expression 5 4 y + 10 y written as a single fraction in its simplest form is 10 9 y .
Examples
Fractions are used in everyday life, such as when calculating proportions in recipes or determining discounts while shopping. For example, if you are mixing paint and need 5 4 of a can of blue paint and 10 1 of a can of yellow paint, you need to know the total fraction of paint you'll be using. Adding these fractions, as we did in the problem, helps you determine that you'll be using 10 9 of a can of paint in total. This ensures you have enough space in your mixing container and helps you accurately measure the required amounts.
To combine 5 4 y + 10 y into a single fraction, find the least common denominator (which is 10), rewrite the first fraction as 10 8 y , and add the two fractions to get 10 9 y , which is already in its simplest form.
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