The sandwich or tortilla law is a method that allows operating with fractions; specifically, it allows you to divide fractions. In other words, through this law you can make divisions of rational numbers. The Sandwich Law is a useful and easy tool to remember.
In this article we will consider only the case of division of rational numbers that are not both integers. These rational numbers are also known as fractional or broken numbers.
Explanation
Suppose you need to divide two fractional numbers a / b ÷ c / d. The sandwich law consists in expressing this division as follows:
This law establishes that the result is obtained by multiplying the number located at the upper end (in this case the number "a") by the number at the lower end (in this case "d"), and dividing this multiplication by the product of the middle numbers (in this case, "b" and "c"). Thus, the above division is equal to a × d / b × c.
It can be seen in the way of expressing the previous division that the middle line is longer than that of the fractional numbers. It is also appreciated that it is similar to a sandwich, since the caps are the fractional numbers that you want to divide.
This division technique is also known as double C, since a large "C" can be used to identify the product of the extreme numbers and a smaller "C" to identify the product of the middle numbers:
Illustration
Fractional or rational numbers are numbers of the form m / n, where "m" and "n" are whole numbers. The multiplicative inverse of a rational number m / n consists of another rational number that, when multiplied by m / n, results in the number one (1).
This multiplicative inverse is denoted by (m / n) -1 and is equal to n / m, since m / n × n / m = m × n / n × m = 1. By notation, we also have that (m / n) -1 = 1 / (m / n).
The mathematical justification of the sandwich law, as well as other existing techniques for dividing fractions, lies in the fact that when dividing two rational numbers a / b and c / d, basically what is being done is the multiplication of a / b by the multiplicative inverse of c / d. This is:
a / b ÷ c / d = a / b × 1 / (c / d) = a / b × (c / d) -1 = a / b × d / c = a × d / b × c, as already had been obtained previously.
In order not to overwork, something that must be taken into account before using the sandwich law is that both fractions are as simplified as possible, since there are cases where it is not necessary to use the law.
For example, 8/2 ÷ 16/4 = 4 ÷ 4 = 1. The sandwich law could have been used, obtaining the same result after simplifying, but the division can also be done directly since the numerators are divisible by the denominators.
Another important thing to consider is that this law can also be used when you need to divide a fractional number by a whole number. In this case, put a 1 under the whole number, and proceed to use the sandwich law as before. This is so because any integer k satisfies that k = k / 1.
Exercises
Here are a number of divisions in which the sandwich law is used:
- 2 ÷ (7/3) = (2/1) ÷ (7/3) = (2 × 3) / (1 × 7) = 6/7.
- 2/4 ÷ 5/6 = 1/2 ÷ 5/6 = 1 × 6/2 × 5 = 6/10 = 3/5.
In this case, the fractions 2/4 and 6/10 were simplified, dividing by 2 up and down. This is a classic method to simplify fractions consisting of finding the common divisors of the numerator and the denominator (if any) and dividing both by the common divisor until obtaining an irreducible fraction (in which there are no common divisors).
- (xy + y) / z ÷ (x + 1) / z 2 = (xy + y) z 2 / z (x + 1) = (x + 1) yz 2 / z (x + 1) = yz.
References
- Almaguer, G. (2002). Mathematics 1. Editorial Limusa.
- Álvarez, J., Jácome, J., López, J., Cruz, E. d., & Tetumo, J. (2007). Basic mathematics, supporting elements. Univ. J. Autónoma de Tabasco.
- Bails, B. (1839). Principles of arithmetic. Printed by Ignacio Cumplido.
- Barker, L. (2011). Leveled Texts for Mathematics: Number and Operations. Teacher Created Materials.
- Barrios, AA (2001). Mathematics 2nd. Editorial Progreso.
- Eguiluz, ML (2000). Fractions: a headache? Noveduc Books.
- García Rua, J., & Martínez Sánchez, JM (1997). Elementary basic mathematics. Ministry of Education.