Question

Evaluate each expression. $$ 2 \cdot \frac{3}{4}+\frac{3}{8} $$

Short answer

\( \frac{15}{8} \)

Step-by-step solution

- Multiply the first term

Firstly, multiply the integer 2 by the fraction \(\frac{3}{4}\). To do this, multiply the numerator by 2: \(\frac{2 \times 3}{4} = \frac{6}{4}\).

- Simplify the fraction

Next, simplify the fraction \(\frac{6}{4}\) by dividing both the numerator and the denominator by their greatest common divisor, which is 2: \(\frac{6 \div 2}{4 \div 2} = \frac{3}{2}\).

- Add the fractions

Now, add the simplified fraction \(\frac{3}{2}\) to the second term \(\frac{3}{8}\). To do this, find a common denominator, which is 8. Convert \(\frac{3}{2}\) to a fraction with a denominator of 8: \(\frac{3 \times 4}{2 \times 4} = \frac{12}{8}\).

- Perform the addition

With both fractions having a common denominator, add them: \(\frac{12}{8} + \frac{3}{8} = \frac{15}{8}\). The result is \(\frac{15}{8}\).

Key concepts

multiplying fractions

Multiplying fractions might sound tricky, but it’s pretty simple. It's all about multiplying the numerators together and the denominators together. For instance, when you need to multiply 2 by \( \frac{3}{4} \), you treat 2 as \( \frac{2}{1} \). So, you multiply the numbers on top (numerators): \( 2 \times 3 \). And then the numbers at the bottom (denominators): 1 \times 4. This gives us \( \frac{6}{4} \). Now, we've performed the multiplication step!

simplifying fractions

Simplifying fractions means making them as simple as possible. When we simplified \( \frac{6}{4} \), we looked for the greatest common divisor (GCD) of 6 and 4. The GCD is the largest number that divides both the numerator and the denominator evenly. In this case, the GCD is 2. So, we divided the numerator and the denominator by 2: \( \frac{6 \div 2}{4 \div 2} = \frac{3}{2} \). Simplifying helps make fractions easier to understand and work with.

common denominator

To add \( \frac{3}{2} \) and \( \frac{3}{8} \), they need a common denominator. This means the bottom numbers (denominators) must be the same. The smallest number both 2 and 8 share is 8. We convert \( \frac{3}{2} \) to \( \frac{12}{8} \) by multiplying both the numerator and the denominator by 4: \( \frac{3 \times 4}{2 \times 4} = \frac{12}{8} \). Now, both fractions have the same denominator. Adding them is simple: \( \frac{12}{8} + \frac{3}{8} = \frac{15}{8} \). Finding a common denominator allows us to add fractions efficiently.