Write $\frac{7 a}{2}-\frac{4}{3}$ as a single fraction in its simplest form.
Answer (2)
Find the least common denominator (LCD) of 2 and 3, which is 6.
Rewrite each fraction with the LCD: 2 7 a = 6 21 a and 3 4 = 6 8 .
Subtract the fractions: 6 21 a − 6 8 = 6 21 a − 8 .
The simplified expression is 6 21 a − 8 .
Explanation
Understanding the Problem We are asked to write the expression 2 7 a − 3 4 as a single fraction in its simplest form. This involves finding a common denominator and combining the two fractions.
Finding the Least Common Denominator To combine the two fractions, we need to find the least common denominator (LCD) of the denominators 2 and 3. The LCD is the smallest multiple that both 2 and 3 share, which is 6.
Rewriting Fractions with Common Denominator Now, we rewrite each fraction with the LCD of 6 as the denominator. To do this, we multiply the numerator and denominator of each fraction by the appropriate factor to get a denominator of 6. For the first fraction, 2 7 a , we multiply both the numerator and the denominator by 3: 2 7 a = 2 × 3 7 a × 3 = 6 21 a . For the second fraction, 3 4 , we multiply both the numerator and the denominator by 2: 3 4 = 3 × 2 4 × 2 = 6 8 .
Subtracting the Fractions Now that both fractions have the same denominator, we can subtract them: 6 21 a − 6 8 = 6 21 a − 8 .
Checking for Simplification The resulting fraction is 6 21 a − 8 . We need to check if this fraction can be simplified further. The numerator is 21 a − 8 , and the denominator is 6. There are no common factors between the numerator and the denominator (other than 1), so the fraction is already in its simplest form.
Final Answer Therefore, the expression 2 7 a − 3 4 written as a single fraction in its simplest form is 6 21 a − 8 .
Examples
Imagine you're baking a cake and a recipe calls for 2 7 cups of flour but you only have 3 4 cups. To figure out how much more flour you need, you would perform the subtraction 2 7 − 3 4 . Expressing this as a single fraction, 6 21 − 8 = 6 13 , tells you exactly how many more cups of flour are needed. This type of calculation is essential in cooking, construction, and many other fields where precise measurements are required.
To write 2 7 a − 3 4 as a single fraction, we find the least common denominator of 2 and 3, which is 6. We rewrite the fractions as 6 21 a and − 6 8 , then combine them to get 6 21 a − 8 . This fraction cannot be simplified further, so the final answer is 6 21 a − 8 .
;
