Find the sum or the difference for each pair of numbers. Show all your work.
For the first problem, we need to add the fractions \(\frac{5}{14}\) and \(\frac{4}{5}\). To add fractions with different denominators, we find a common denominator. The least common multiple (LCM) of 14 and 5 is 70.
To convert \(\frac{5}{14}\), we multiply the numerator and the denominator by 5:
\[\frac{5}{14} = \frac{5 \times 5}{14 \times 5} = \frac{25}{70}\]
To convert \(\frac{4}{5}\), we multiply the numerator and the denominator by 14:
\[\frac{4}{5} = \frac{4 \times 14}{5 \times 14} = \frac{56}{70}\]
Now we add the two fractions together:
\[\frac{25}{70} + \frac{56}{70} = \frac{25 + 56}{70} = \frac{81}{70}\]
The sum is \(\frac{81}{70}\).
For the second problem, we need to compute the difference \(\frac{2}{9} - \frac{4}{7}\) using the provided format \(\frac{?}{63} - \frac{?}{63} = \frac{?}{63}\).
First, we express \(\frac{2}{9}\) with a denominator of 63:
\[\frac{2}{9} = \frac{2 \times 7}{9 \times 7} = \frac{14}{63}\]
Next, we express \(\frac{4}{7}\) with a denominator of 63:
\[\frac{4}{7} = \frac{4 \times 9}{7 \times 9} = \frac{36}{63}\]
Now we perform the subtraction:
\[\frac{14}{63} - \frac{36}{63} = \frac{14 - 36}{63} = \frac{-22}{63}\]
The difference is \(\frac{-22}{63}\).
When splitting tasks or resources, such as mixing ingredients for a recipe or dividing land, you often deal with fractions of different sizes. By finding a common denominator, you convert these parts into the same unit, allowing for precise addition or subtraction. This ensures accuracy in fields ranging from cooking and carpentry to financial planning and engineering.