NCERT 5th Class (CBSE) Mathematics: Fractions

A faction shows a part of a whole. A whole can be region or a collection. You are familiar with these fractions. Like Fractions: Fractions that have same denominator are called like fractions, e.g., 3/7, 5/7, 1/7 Unlike fractions: Factions that have different denominators are called unlike fractions, e.g., 4/7, 3/8, 2/3 Proper fractions: A fractions with the numerator less than the denominator is a proper fraction. A proper fraction has a value less than 1. For example, 1/8, 4/9, 3/5 Improper fraction: A fraction that has the numerator greater that equal to the denominator is called an improper fraction and its value is 1 or more than 1. For example, 5/2, 10/7, 8/3, 9/9 Mixed number: When we combine a whole number and a fraction, we get a mixed number. For example, Equivalent fractions: Fractions that name the same part are called equivalent fractions. You can find equivalent fractions by multiplying or dividing the numerator and the denominator by the same number. 1 (×4)/2 (×4) = 4/8 4 (÷2)/8 (÷2) = 2 (÷2)/4 (÷2) = 1/2 You can cross multiply to check whether two fractions are equivalent. Fraction in the lowest terms: A fraction is in its lowest term when the only common factor between the numerator and the denominator is 1. 1/2, 2/4, 3/6 are all equivalent fractions, but 1/2 is in its lowest term. To reduce a fraction to its lowest terms you keep dividing the numerator and the denominator by the same number till you cannot divide further. 18/30 = 18 (÷2)/30 (÷2) = 9/15 = 9 (÷3)/15 (÷3) = 3/5 Addition of fractions Like fractions 3/5 + 1/5 = 4/5 Unlike fractions 2/5 + 1/4 = 8/20 + 5/20 = 13/20 LCM of 5 and 4 is 20 Mixed numbers Subtraction of fractions Like fractions 8/9 - 3/9 = 5/9 Unlike fractions 5/6 - 3/5 = 25/30 - 18/30 = 7/30 LCM of 6 and 5 is 30 Mixed numbers MULTIPLICATION OF FRACTIONS A. Multiplying Fractions by whole Numbers (a) Shalini is marking tea six people. She needs 2/3 teaspoon of leaves for each cup. How much will she need for 6 cups? Repeated addion can also be also be shown as multiplication. 6×2/3 = ? Step 1. Write the whole number as fraction Step 2. Multiply the numerators Step 3. Multiply the denominators Step 4. Reduce to the lowest term 6×2/3 = 4 Answer: Shalini needs 4 teaspoons of tea leaves to make 6 cups of tea. (b) 5×3/8 = ? 5×3/8 = 5/1×3/8 5/1×3/8 = 15/8 (c) 5/6×1 = ? 5/6×1 = 5 (d) 0×3/7 = ? 0×3/7 = 0 (e) In a class of 40 children, 3/4 use the school bus to come to school. How many children use the school bus? To find 3/4 of 40, you can multiply. 3/4×40 ? 3/4×40 = 3/4×40/1 = 120/4 = 30 Answer: 30 students in the class come to school by the school bus. (f) What is 2/3 of 4 meters B. Multiplying Fractions by Fractions (a) This is Venkatesh's plot land. He wants to build a house on it. He wants to keep 1/5 of the land as a garden in front of the house. Of the portion meant for the house, he keeps 1/4 as garage space for his car. The rest of the space is for building the house. What part of the total land does Venkatesh use to build the house? From the diagram we can see that Venkatesh uses 12/20 part of the total land to build his house. To be able to find this answer without a diagram, we must solve it as follows. Venkatesh uses 3/4 of 4/5 of the total land to build his house. 3/4 of 4/5 = ? 3/4 of 4/5 is the same as 3/4×4/5 3/4×4/5 = ? Step 1. Multiply the numerators: 3/4×4/5 = 12 Step 2. Multiply the denominators: 3/4×4/5 = 12/20 Step 3. Reduce to the lowest term: 3/4×4/5 = 12/20 = 3/5 Venkatesh uses 3/5 of the land to build his house. (b) 2/3×1/7 = 2/21 1/7×2/3 = 2/21 (c) 3/4×7/15 = 21/60 = 7/20 or 3/4×7/15 Step 1. Find a common factor between any numerator and the denominator in the given question: 3/4×7/15 (3 and 15 have the common factor 3) Step 2. Divide that numerator and the denominator by the common factor and write the quotient. Step 3. Multiply the new numerators and denominators. Step 4. Write in lowest terms if not already in lowest term = 7/20 (d) 9/25×10/18 = ? 9/25×10/18 = 1/5 C. Multiplying More Than Two Fractions When you multiply more than two fractions, you can multiply any two fractions first and then multiply the product with the remaining fractions. 3/4×5/6×4/15 D. Multiplying Mixed Numbers Step 1. Write the mixed number as an improper fraction Step 2. Multiply the numerators. Step 3. Multiply the denominators. Step 4. Give the answer as a mixed number RECIPROCALS Two numbers whose product is 1 are called reciprocals. For example, 5/6 and 6/5 are reciprocals. 5/6×6/5 = 30/30 = 1 4 and 1/4 are reciprocals. Step 1. Write the number as a fraction 7/1, 3/4, 5/2 Step 2. Interchange the numerator and denominator 1/7, 4/3, 2/5 DIVISION OF FRACTIONS Dividing Fractions To understand division of fractions let us look at division again. If you had a sum 6÷2, you could think of it as 'how many 2s are there in 6?' The answer would be 3. There are three 2s in 6. Dividing a Whole Number by an Unit Fraction (a) You can do the same thing when you divide by fractions. To understand 3÷1/2, ask the question 'how many halves in 3?' These are three apples. Think: 'how many halves in three apples?' We can see that there are 6 halves in 3 apples. 3÷1/2 = 6 We can also describe the same picture as 3×2 = 6 Dividing by 1/2 is just like multiplying by 2. Here is how you can solve the sum without pictures. 3 ÷ 1/2 = ? Dividend Divisor Step 1. Find the reciprocal of the divisor. Reciprocal of 1/2 is 2/1. Step 2. Multiply the dividend with the reciprocal: 3×2/1 = 6 3÷1/2 = 6 Dividing a Whole Number by a Fraction (b) 3÷3/4 = ? Think: How many 3/4 in 3? Let us understand this with the help of an example. Here are three pizzas. How many 3/4 ths in three pizzas? First cut each pizza into 1/4th pieces. Then separate them into 3/4ths. Answer: There are four 3/4 ths in 3 pizzas. Here is how you can solve this without pictures. 3÷3/4 = ? Step 1. Find the reciprocal of the divisor. Reciprocal of 3/4 is 4/3 Step 2. Multiply the dividend with the reciprocal: 3×4/3 = 4 Dividing a Fraction by a Whole Number (c) Ajay has 3/4 l juice. If he pours it equally into 4 glasses, how much will each glass hold? 3/4÷4 = ? 3/4÷4 = 3/4÷4/1 = 3/4×i/4 = 3/16 Answer: Each glass will hold 3/16 l juice. (d) 5/7÷1 = ? 5/7÷1/1 = 5/7×1/1 =5/7 5/7÷1 = 5/7 Dividing a Fraction by a Fraction USING A NUMBER LINE You can use a number line to show division of a whole number by a fraction. (a) 5÷2/3 = ? 5÷1/3 = 15 (b) 2÷2/5 = ? Number line showing 5 equal parts for each number. There are 5 two-fifths in 2. 2÷2/5 = 5 Dividing with Mixed Numbers [ads]