Fraction Operations Quiz

Study Notes

Adding Fractions

Adding fractions is similar to adding whole numbers. To add two fractions, you need to have like denominators. If the denominators are not the same, then you will need to find the lowest common multiple (LCM) of both denominators and change each fraction so they have the LCM denominator. For example, to add \(\frac{2}{7}\) and \(\frac{5}{8}\), we first find the LCM of 7 and 8 which is 28. Then we change \(\frac{2}{7}\) to \(\frac{2 \times 4}{7 \times 4} = \frac{8}{28}\) and \(\frac{5}{8}\) to \(\frac{5 \times 3}{8 \times 3} = \frac{15}{24}\). Now the denominators are the same, 28, and the numerator sums are equal, (8 + 15 = 23), so the result is \(\frac{23}{28}\).

Subtracting Fractions

Subtracting fractions is also similar to subtracting whole numbers. To subtract one fraction from another, you need the same numerical value in their denominators. This means finding the greatest common divisor (GCD) of the denominators and adjusting each fraction's denominator accordingly. For example, to subtract \(\frac{3}{4}\) from \(\frac{6}{7}\), we first find the GCD of 4 and 7, which is 2. We then divide both the numerator and denominator by 2 (since it's the highest power that divides them): \(\frac{3}{4}\) becomes \(\frac{\frac{1}{2}}{1}\) and \(\frac{6}{7}\) becomes \(\frac{\frac{3}{2}}{1}\). Now the denominators are the same (in this case, just 1), and the difference of the numerators is equal ((6 - 3 = 3)), so the result is \(\frac{\frac{3}{2}}{1} - \frac{\frac{1}{2}}{1} = \frac{2}{1}\).

Multiplying Fractions

Multiplying fractions involves multiplying the numerators and then multiplying the denominators. The resulting product has its numerator divided by the product of the original denominators. For example, if we multiply \(\frac{2}{5}\) by \(\frac{3}{7}\), we get \(\frac{2 \times 3}{5 \times 7} = \frac{6}{35}\).

Fraction Simplification

Fraction simplification is the process of reducing a fraction to its simplest form by dividing both its numerator and its denominator by their greatest common factor (GCF). For example, if we have \(\frac{10}{15}\), we can simplify it by dividing both the numerator and the denominator by their GCF, which is 5: \(\frac{10}{15} => \frac{10 / 5}{15 / 5} = \frac{2}{3}\). This gives us the simplified fraction \(\frac{2}{3}\).

Dividing Fractions

Dividing fractions involves flipping the second fraction and multiplying the first fraction by the reciprocal of the second fraction. Since finding the reciprocal of a fraction changes its position (i.e., swapping numerator and denominator), it results in a different fraction being multiplied with the original division problem. For example, to divide \(\frac{3}{4}\) by \(\frac{2}{3}\), we flip \(\frac{2}{3}\) to get \(\frac{3}{2}\) and then multiply \(\frac{3}{4}\) by \(\frac{3}{2}\): \(\frac{3}{4} \times \frac{3}{2} = \frac{9}{8}\).