Working with Fractions · Lesson 1 of 8
Multiplying Fractions and Whole Numbers
“Understand repeated fractional amounts and find a fraction of a whole-number quantity.”
• Interpret a whole number times a fraction as repeated addition. • Find a fraction of a quantity by sharing first and then taking the required shares. • Use mixed numbers in multiplication problems. • Connect an answer to its unit and express it in a suitable form.
Multiplication of Fractions
Suppose a person walks 3 kilometres in one hour. At the same steady pace, five hours gives five copies of 3 kilometres, or 15 kilometres. Multiplication connects a repeated amount with the number of times it is taken. That idea still works when an amount, or the number of times we take it, is a fraction.
A fraction describes a quantity using equal parts of a chosen whole. In 3/5, the denominator 5 tells us how many equal parts make one whole; the numerator 3 tells us how many of those parts are taken. Before calculating, identify the whole: it might be one kilometre, one litre, one hour, or one complete piece of land.
A Whole Number Times a Fraction
Imagine a tortoise that covers 1/4 kilometre in each hour. Three hours gives three copies of that distance. Since every copy is measured in quarters of a kilometre, add the numerators while keeping the quarter-sized unit unchanged.
Problem
A tortoise covers 1/4 km each hour. How far does it walk in 3 hours?
- 1.There are 3 equal one-hour periods, each contributing 1/4 km.
- 2.Add the three distances: 1/4 + 1/4 + 1/4 = 3/4.
- 3.Therefore 3 × 1/4 = 3/4. The distance is 3/4 km.
More than one whole can result from repeated fractional amounts. If five grandchildren each receive 2/3 acre, count ten one-third-acre shares. Ten thirds is 10/3 acres, or 3 1/3 acres. The denominator stays 3 because the pieces are still thirds of an acre.
Problem
A farmer gives 2/3 acre to each of five grandchildren. How much land does she give altogether?
- 1.Multiply the number of recipients by the amount per recipient: 5 × 2/3.
- 2.Five copies of two thirds make 10 thirds, so the result is 10/3 acres.
- 3.Divide 10 by 3: three whole acres use 9 thirds and leave 1 third. The total is 3 1/3 acres.
A Fraction of a Whole Number
Now suppose the walker travels for 2/5 of an hour. We cannot interpret this as two whole hours repeated five times. Instead, first find what happens in one fifth of an hour, and then take two of those equal intervals. The same sharing-and-taking idea explains finding any fraction of a quantity.
Problem
Aaron walks 3 km in 1 hour at a steady pace. How far does he walk in 2/5 hour?
- 1.Split the hour into 5 equal periods. The distance in each period is 3 ÷ 5 = 3/5 km.
- 2.Two fifths of an hour contains 2 of these periods. The distance is 2 × 3/5 = 6/5 km.
- 3.Thus 2/5 × 3 = 6/5. Expressed as a mixed number, the distance is 1 1/5 km.
The order of these two steps matters to understanding: divide the original quantity into the denominator’s number of equal parts, then take the numerator’s number of parts. For 2/5 of 3, we divide 3 by 5 and take two shares. This is the meaning of multiplying by 2/5, even though later we can calculate the product more quickly.
Mixed Numbers and Rates
A mixed number combines whole units with a remaining fraction. Converting it to one fraction makes it easier to multiply. For 1 1/4, the whole contains 4 quarters, so there are 5 quarters altogether. Keep track of what the given rate means: cost per hour multiplied by hours gives total cost.
Problem
Internet time costs ₹8 per hour. What is the cost for 1 1/4 hours?
- 1.Convert 1 1/4 hours to 5/4 hours because one whole hour is 4/4 hour.
- 2.One quarter hour costs 8 ÷ 4 = ₹2.
- 3.Five quarter hours cost 5 × ₹2 = ₹10. Equivalently, 5/4 × 8 = 40/4 = 10.
For 7 × 3/5, do not change the denominator to 7 × 5. You are taking seven copies of a three-fifths quantity; the pieces remain fifths. Also, 3/5 of a quantity means divide by 5 and take 3 shares, not divide by 3 and take 5 shares.
Draw a strip for 4 × 1/3. Explain why the shaded amount is greater than one whole even though each individual amount is less than one whole.
Quiz
What does 6 × 1/4 mean?
What is 2/3 of 12?
Which pair of steps finds 3/7 of 14?
What is 5 × 2/3 as a mixed number?
A service costs ₹12 per hour. What is the charge for 1 1/2 hours?
Practice Problems
- Tenzin drinks 1/2 glass of milk daily. Find the amount for one week and for January, which has 31 days.
- Workers build 1 km of canal in 8 days at a steady rate. Find the length built in one day and in a five-day working week.
- Three families share 5 litres of oil equally each week. Find the amount per family for one week and for four weeks.
- The Moon sets 5/6 hour later each day in a simplified model. Starting from Monday at 10 pm, how many hours later is Thursday’s setting time, and what clock time is that?
- Calculate and convert to mixed numbers: 7 × 3/5; 4 × 1/3; 9/7 × 6; 13/11 × 6.
- A cyclist covers 10 km each hour. Find the distance for 3/4 hour and explain the two sharing-and-taking steps.
Milk: 7/2 = 3 1/2 glasses and 31/2 = 15 1/2 glasses. Canal: 1/8 km and 5/8 km. Oil: 5/3 litres and 20/3 litres. Moon: three daily intervals give 3 × 5/6 = 5/2 hours; The cumulative delay for the Thursday-associated setting is 2 1/2 hours, placing it at 12:30 am after Thursday evening, on Friday. Here the day labels refer to successive nightly setting events. Products: 4 1/5; 1 1/3; 7 5/7; 7 1/11. Cycling: 10 ÷ 4 = 5/2 km per quarter hour; take three shares to get 15/2 = 7 1/2 km.
Key Takeaways
• A whole number times a fraction counts repeated amounts of the same fractional unit. • To find a/b of a quantity, divide into b equal shares and take a shares. • A product can include several whole units; convert improper fractions to mixed numbers when useful. • Convert a mixed number to an improper fraction before using a single multiplication calculation. • Attach the correct unit to the result and connect it to the original situation.
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Multiplying Two Fractions