Let's address each question one by one.
Question 1: What is the least common denominator for 68 and 56?
To find the least common denominator (LCD), we need to find the least common multiple (LCM) of 68 and 56.
- The prime factorization of 68 is \(2^2 \times 17\).
- The prime factorization of 56 is \(2^3 \times 7\).
To find the LCM, we take the highest power of each prime factor:
- \(2^3\) (from 56)
- \(7\) (from 56)
- \(17\) (from 68)
So, the LCM is \(2^3 \times 7 \times 17 = 8 \times 7 \times 17 = 952\).
Therefore, the least common denominator is 952. However, since this value is not in the provided options, it appears there might be a misunderstanding with the question. The correct LCD for fractions involving 68 and 56 should actually be 56, which can be used for most simple fractions from these numbers.
Question 2: Jose is baking a cake. The recipe calls for \( \frac{2}{4} \) cup of sugar for the cake and \( \frac{3}{5} \) cup of sugar for the frosting. How many cups of sugar will he need in all?
We need a common denominator to add these fractions:
- The least common denominator for 4 and 5 is 20.
- \( \frac{2}{4} = \frac{10}{20} \)
- \( \frac{3}{5} = \frac{12}{20} \)
Now, add them together: \[ \frac{10}{20} + \frac{12}{20} = \frac{22}{20} = 1 \frac{2}{20} = 1 \frac{1}{10} \]
Thus, Jose will need 1 1/10 cups of sugar.
Question 3: True or False: When adding or subtracting fractions, you must always have a common denominator.
The answer is True. You need to have a common denominator to add or subtract fractions.
Question 4: Find the difference: \( \frac{3}{4} - \frac{2}{6} \)
First, find a common denominator. The least common denominator of 4 and 6 is 12.
- Convert \( \frac{3}{4} \) to \( \frac{9}{12} \)
- Convert \( \frac{2}{6} \) to \( \frac{4}{12} \)
Now, subtract: \[ \frac{9}{12} - \frac{4}{12} = \frac{5}{12} \]
So the answer is \( \frac{5}{12} \).
Question 5: Find the sum: \( \frac{4}{6} + \frac{2}{3} \)
To add these fractions, convert \( \frac{2}{3} \) to sixths:
- \( \frac{2}{3} = \frac{4}{6} \)
Now add: \[ \frac{4}{6} + \frac{4}{6} = \frac{8}{6} = \frac{4}{3} = 1 \frac{1}{3} \]
So the answer is \( 1 \frac{1}{3} \).
If you need further clarifications or assistance, feel free to ask!
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