two denominators (bottom number) of each fraction. In case of an integer number apply the invisible denominator For adding quarters to thirds, both types of fraction are converted to twelfths, thus: Consider adding the following two quantities: First, convert into fifteenths by multiplying both the numerator and denominator by three: . Since equals 1, multiplication by does not change the value of the fraction. Second, convert into fifteenths by multiplying both the numerator and denominator by five: . Now it can be seen that: is equivalent to: This method can be expressed algebraically: This algebraic method always works, thereby guaranteeing that the sum of simple fractions is always again a simple fraction. However, if the single denominators contain a common factor, a smaller denominator than the product of these can be used. For example, when adding and the single denominators have a common factor and therefore, instead of the denominator 24 (4 × 6), the halved denominator 12 may be used, not only reducing the denominator in the result, but also the factors in the numerator. The smallest possible denominator is given by the least common multiple of the single denominators, which results from dividing the rote multiple by all common factors of the single denominators. This is called the least common denominator. Subtraction The process for subtracting fractions is, in essence, the same as that of adding them: find a common denominator, and change each fraction to an equivalent fraction with the chosen common denominator. The resulting fraction will have that denominator, and its numerator will be the result of subtracting the numerators of the original fractions. For instance, Multiplication Multiplying a fraction by another fraction To multiply fractions, multiply the numerators and multiply the denominators. Thus: To explain the process, consider one third of one quarter. Using the example of a cake, if three small slices of equal size make up a quarter, and four quarters make up a whole, twelve of these small, equal slices make up a whole. Therefore, a third of a quarter is a twelfth. Now consider the numerators. The first fraction, two thirds, is twice as large as one third. Since one third of a quarter is one twelfth, two thirds of a quarter is two twelfth. The second fraction, three quarters, is three times as large as one quarter, so two thirds of three quarters is three times as large as two