Division: Fractions
Division involving fractions is a fundamental mathematical operation that extends from basic division concepts. At its core, division is the process of splitting a quantity into equal parts, and it utilizes specific terminology: the dividend (the number being divided), the divisor (the number by which the dividend is divided), and the quotient (the result of the division). To effectively work with fractions, one must learn to use algorithms that facilitate operations, especially as numbers become larger.
When dividing a fraction by a whole number, it is essential to convert the whole number into a fraction, find its reciprocal, and multiply the two fractions together. For example, to divide 2/3 by 8, you convert 8 to 8/1, find its reciprocal (1/8), and then multiply: (1/8) × (2/3) results in 2/24, which simplifies to 1/12. Conversely, when dividing whole numbers by fractions, the process involves multiplying by the reciprocal of the fraction. This method can seem complex at first, but understanding the underlying concepts simplifies the process significantly.
Additionally, when dealing with negative fractions, it is important to focus on the absolute values first and determine the sign of the final answer based on the number of negative signs encountered. Practical applications of fraction division can be illustrated with real-world scenarios, such as calculating resources needed for projects or determining time allocations for study. By mastering these division techniques, learners can enhance their mathematical fluency and problem-solving skills.
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Division: Fractions
As with addition, subtraction, and multiplication, progress is made by learning algorithms that allow the performance of operations beyond basic facts. After learning these basic division facts (such as division is the process of splitting into equal parts) and the concept of division, algorithms can be introduced that will allow the division of larger numbers. It is important to understand that it is necessary to learn how to use algorithms to divide larger numbers. One should also understand the vocabulary: dividend, divisor, and quotient. These three terms are related by the following equation: dividend ÷ divisor = quotient.
The dividend is the numerator: the number that is being divided. The divisor is the denominator: the number one divides by. Divisor can also mean a number that divides an integer evenly (i.e., with no remainder). For example, in 20/5 = 4, 20 is the dividend and 5 is the divisor. Finally, a quotient is simply the result of dividing a numerator by a denominator.
Division is the opposite of multiplication. When one knows a multiplication fact, one can find the complementary division fact. For example, 2 × 7 = 14, so 14 ÷ 7 = 2 and also 14/2 = 7. Dividing a fraction by a whole number is straightforward. To divide a fraction by a whole number, convert the whole number into a fraction, find the reciprocal of that fraction, and multiply the result by the first fraction—then simplify. For example, take 2/3 ÷ 8. Convert the whole number (here 8) into a fraction (so 8/1). Find the reciprocal: 1/8. Multiply the result by the first fraction (1/8 × 2/3), which comes to 2/24. This can easily be simplified: Find the lowest common denominator, and divide both the numerator and denominator by any number that divides evenly into both numbers. Since 2 is the numerator, you should see if 2 divides evenly into 24 (it does because 24 is even). Then, divide both the numerator and denominator by 2 to get the new numerator and denominator for a simplified answer: 1/12.
Overview
Dividing whole numbers by fractions can best be approached by multiplying the denominator of the fraction by the whole number. Having done so, all that remains is to simplify the resulting fraction, if necessary. To divide by fractions, it is necessary to invert the divisor and multiply. To invert a fraction, reverse the numerator and denominator. This inverted fraction is called the reciprocal. The reciprocal of 3/4 is 4/3. To find the reciprocal of a mixed number, first convert the mixed number into its improper fraction (e.g., 1 3/5 gives 8/5 so the reciprocal of 1 3/5 is 5/8). To divide a number by a fraction, first multiply the number by the reciprocal of the fraction. In other words, when the operation is changed from ÷ to ×, change the second fraction to its reciprocal. For example, 3 1/2 divided by 3/5 goes to 7/2 ÷ 3/5, goes to 7/2 × 5/3, goes to (7 × 5) / (2 × 3); hence, 35/6. Simplify to 5 5/6.
It may initially seem that dividing fractions is difficult. This is not the case, however, once the underlying mathematical concepts are understood. Remember the following simple rules to divide any negative fraction problem encountered.
First, ignore the negative sign and multiply one fraction by the reciprocal of the other, as in the examples above. Remember that the reciprocal is the result of making the numerator the denominator, and vice versa. For example, the reciprocal of 3/7 would be 7/3. Next, simplify the new fraction, reducing it as needed. For instance, if your multiplication result was 8/4, your answer simplifies to 2. Finally, determine whether the final resulting fraction becomes negative or positive by looking at the number of negative signs in the process. An even number of negative signs results in a positive answer and an odd number results in a negative answer.
Dividing fractions by fractions and whole numbers applications is best understood with an example. At her favorite store, Jane has found beautiful balls of wool, with 10% off the price. She can make one dress from 6/7 of a ball of wool. If Jane buys 18 balls of wool, how many dresses can she make? To find out, divide the total number of balls of wool by the quantity needed to make one dress (6/7 of a ball of wool). This gives (18 balls of wool) / (6/7 balls per dress). As above, dividing by 6/7 is achieved by multiplying by its reciprocal (i.e., 7/6). So the total number of dresses that Jane can make is given by 18 × 7/6 (i.e., 21 dresses). Also, Jane can purchase the 18 balls of wool at a discount price: 10% off the total. Say the price per ball before discounting is $3.00. After discounting, $2.70. Eighteen balls cost $48.60. Price per dress is $2.31.
Division with fractions and whole numbers word problems is best illustrated by way of an example. Alice will spend 2/3 of this weekend revising for five examinations. If she spends the same amount of time on each examination, what fraction of the time available will she spend revising for each examination?
To solve, divide the amount of time available (2/3 of this weekend) by how many examinations she will study for (5). In other words, calculate 2/3 ÷ 5, which gives 2/15 of this weekend per examination. Alice will spend 2/15 of this weekend revising each examination. (Note how the problem and its solution are expressed in words, rather than in mathematical symbols and operators.)
Bibliography
Alcock, L. How to Study for a Mathematics Degree. New York: Oxford UP, 2012.
Houston, Kevin. How to Think Like a Mathematician: A Companion to Undergraduate Mathematics. New York: Cambridge UP, 2009.
Liebeck, Martin. A Concise Introduction to Pure Mathematics. 3rd ed. \Chapman Hall/CRC, 2010.
Ni, Yujing; and Yong-Di Zhou. "Teaching and Learning Fraction and Rational Numbers: The Origins and Implications of Whole Number Bias." Educational Psychologist 40.1 (2005).