Before we get to the point of how we can solve various types of fraction word problems, let’s see what fractions are, kinds of fractions, as well as what you can do with them.

Fraction in mathematics is a number consisting of one or more parts (shares) of a unit. Fractions are part of the field of rational numbers. By way of the recording fractions, they are divided into two formats: ordinary and decimal.

The term fraction comes from the Latin «fractura», which, in turn, is a translation of the Arabic term with the same meaning: «to break» or «break in pieces». The foundation of the theory of ordinary fractions was laid by Greek and Indian mathematicians. Through Arabs the term came to Europe, where it was already mentioned in the Fibonacci. The terms of the numerator and denominator were invented by the Greek mathematician Maximus Planudes.

Initially, European mathematicians operated only with common fractions, and in astronomy – with sexagesimal. The current designation of fractions is derived from ancient India. At first, the fractions didn’t use slashes. The application of slashes was made permanent only about 300 years ago. In Europe, the first scientist who used and distributed Indian system of account, including the method of recording fractions, was the Italian merchant and a traveler, Fibonacci. A complete theory of fractions and fraction word problems was developed in the 16^{th} century.

Decimal fractions are first found in China around the 3rd century BC among the calculations on the counting board. In written sources decimals were represented in the traditional format, but it gradually replaced the traditional position system. Persian mathematician and astronomer Al-Kashi in his treatise «The key of arithmetic» declared himself the inventor of decimal fractions.

In Europe, the first decimals were introduced by Immanuel Bonfis around 1350, but they became widespread only after the appearance of the works of Simon Stevin «The Tenth». Stevin wrote decimal fractions in complex ways. Commas to separate the integer part began to be used in the 17^{th} century.

Types of Fractions

- Ordinary fractions.

Ordinary or simple fraction is a record of a rational number in the form m/n, where n isn’t equal to 0. The horizontal line (which can be vertical as well) denotes the division sign, resulting in a quotient. The dividend is called the numerator of a fraction and the divisor is the denominator. There are several types for recording ordinary fractions: 1/2 (the slash is called the «solidus»); through the horizontal line, and the lowercase formula. - Proper and improper fractions.

Proper fraction is a fraction, whose numerator module is less than the denominator module. The improper fraction is a rational number, which by the module is bigger or equal than one. - Mixed fractions.

The fraction recorded as a whole number and a proper fraction is called a mixed fraction and is understood as the sum of this number and the fraction. Any rational number can be written as a mixed fraction. In contrast to the mixed fraction, the fraction, containing only the numerator and denominator, is called simple. - Composite fractions.

Composite fraction is an expression, containing multiple horizontal (or inclined) features. - Decimal fractions.

Decimal fraction is a positional record of fractions, for example, 3.24587. The part of the recording, which is before the positional comma, is the integral part of a number (fraction), and he part standing after the comma is a fractional part. Every ordinary fraction can be converted to the decimal fraction, which in this case has a finite number of digits after the decimal point, or is a periodic fraction. Generally speaking, for the positional record of a number it is possible to use not only decimal system, but also other systems (including Fibonacci).

In order to solve different fraction word problems, you need to know what actions you can do with fractions. So, what can you do with fractions? Actually, anything that you can do with the usual numbers: subtract, add, multiply, and divide.

In many fraction word problems you will need to add or subtract fractions. If there is the same number in both fractions’ denominators, then in order to add the fractions, it is necessary to summarize their numerators, and in order to subtract the fractions, you have to subtract their numerators (in the same order). The resulting sum or difference will be the numerator of the result; the denominator will remain the same. If there are different numbers in the denominators of fractions, you must first bring fractions to a common denominator. When adding the mixed numbers, their integers and fractional parts are added separately. When subtracting the mixed numbers, it is recommended to first convert the fractions to the improper fractions, and then subtract one from the other, and then re-cast the result, if required, to the form of the mixed number.

Multiplication is also often needed when dealing with fraction word problems. Multiplying a number by a fraction means to multiply it by the numerator and split the found value by the denominator. Consequently, we have a general rule of multiplication of fractions: to multiply fractions, multiply singly their numerators and denominators and divide the first product by the second.

Fraction word problems can also be about dividing fractions. In order to split some number by a fraction, multiply this number by the inverse fraction. This rule stems from the definition of division.

All students need to know how to solve basic fraction word problems, i.e. be able to find the part of a number and the number on the part. There are three main types of fraction word problems that help understand the meaning of the fraction.

The fraction word problems are always about a certain value A, which is taken as a unit («the whole»), and some part of B, which is expressed in the following fraction: m/n.

The type of fraction word problems is determined by what is unknown – A, B, or m/n. Accordingly, there are three types of fraction word problems:

- Fraction word problems for finding a part of the number, which is expressed in a fraction.

1 – A; m/n - ? In order to find the part of a number, expressed as a fraction, that number can be divided by the denominator of the fraction and multiplied by its numerator: B = A / n x m. - Fraction word problems for finding the number by its part, expressed in a fraction.

1-?; m/n – B. In order to find the number by its part, expressed as a fraction, it is possible to divide this part by the numerator of a fraction and multiply it by the denominator: A = B / m x n. - Fraction word problems for finding the fraction, which one number is from the other number.

1 – A; ? – B. To find a fraction, which one number is from the other number, it is possible to divide the first number by the second number: m/n = A / B.

When studying fractions, for successful solving of fraction word problems it is important for students to understand what is taken as one (whole) in each problem, for how many parts it is broken, what the value of one share is, how many parts are taken, what is the meaning of all the parts combined, what the rules for finding a fraction of the number are, as well as the rules for finding the number by the fraction, and fractions, which is composed by the number from the other number.

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