Working with Fractions | CBSE Class 7 Maths Notes

This note covers multiplication of fractions and whole numbers, fractional parts of quantities, area models, cancelling common factors, comparisons of products, reciprocals, division of fractions, equal sharing, combined filling rates, shaded areas and problems involving successive fractional parts.

How does multiplication help us work with fractions?

A whole number is one of the numbers 0, 1, 2, 3 and so on. A fraction represents a number using equal parts of a whole. In the notation a/b, a is the numerator, which counts the parts, and b is the denominator, which tells how many equal parts make one whole. Here a and b represent whole numbers, with b not zero.

A unit fraction has numerator 1. Thus, 1/4 means one of four equal parts, while 3/4 means three such parts.

The symbols used below are + for addition, − for subtraction, × for multiplication, ÷ for division and = for equality. A slash separates a fraction's numerator and denominator; brackets group quantities to be treated together.

What do the numbers in a multiplication mean?

The product is the result of multiplication. The numbers being multiplied are called factors. In 2/5 × 3, the multiplier is 2/5 and the multiplicand is 3. The multiplier tells what multiple or fractional part of the multiplicand to take.

When a whole number multiplies a fraction, multiplication can be understood as repeated addition. Aaron's pet tortoise travels 1/4 kilometre in one hour. The abbreviation km means kilometre, a unit of distance.

Worked example 1. A tortoise covers 1/4 km in one hour. At the same pace, how far does it travel in three hours?

Answer: Add three equal hourly distances: 1/4 + 1/4 + 1/4 = 3/4. Equivalently, 3 × 1/4 = 3/4. The tortoise travels 3/4 km.

Each term in the addition refers to one hour's travel. The denominator stays 4 because the pieces being added are still quarters of a kilometre. The numerator becomes 3 because three of these pieces have been counted.

How do we find a fractional part of a whole number?

Multiplication by a fraction asks us to take a fractional part of a quantity. To find 2/5 of a quantity, first divide it into five equal parts, then take two of those parts. The word “of” here corresponds to multiplication.

Aaron walks 3 km in one hour. At that pace, the distance in 1/5 hour is one fifth of 3 km. Divide the distance into five equal parts to obtain 3/5 km. Two fifths of an hour gives twice this distance.

Worked example 2. Aaron covers 3 km in one hour. How far does he walk in 2/5 hour at the same pace?

Answer: In 1/5 hour he covers 3 ÷ 5 = 3/5 km. In 2/5 hour he covers 2 × 3/5 = 6/5 km. Thus, 2/5 × 3 = 6/5.

How do mixed fractions enter a calculation?

A rate expresses one quantity for a given amount of another, such as cost per hour. A mixed fraction combines a whole-number part with a fractional part. For example, 1 1/4 means 1 + 1/4. Since one whole contains four quarters, 1 1/4 is the same as 5/4.

Worked example 3. Internet time costs ₹8 for one hour. The symbol ₹ denotes rupees. Find the cost of 1 1/4 hours at this rate.

Answer: Convert 1 1/4 to 5/4. The cost is 5/4 × 8 = (5 × 8)/4 = 5 × 2 = 10. Therefore, the internet time costs ₹10.

The fraction need not be smaller than one. Taking 5/4 of the hourly cost means taking five quarter-hour shares. The whole hour and the additional quarter-hour are both included in the multiplication.

A farmer distributing 2/3 acre to each of five grandchildren uses the same repeated-addition idea. Area measures the surface covered by a shape. An acre is a unit of land area. The total land given is 5 × 2/3 = 10/3 acres.

How does a unit square show multiplication of two fractions?

A unit square has side length one unit and area one square unit. Treating the square as one whole allows us to see a fractional part of another fractional part.

To find 1/2 × 1/4, first take one quarter of the square. Divide that quarter into two equal parts. Each resulting part is one eighth of the whole square, so taking half of one quarter gives 1/8.

How can rows and columns show the answer?

Consider 3/4 × 2/5. Divide the whole square into five equal horizontal rows. Two rows represent 2/5. Divide the square into four equal vertical columns. It now contains twenty equal small rectangles.

The square has five rows and four columns. Its top two rows are yellow. The two cells in the leftmost column of these rows are hatched, representing 2/20 of the whole.

See Fig. 8.1 in your NCERT textbook

The hatched part is one quarter of the two yellow rows. Taking three such quarters gives six small rectangles out of twenty. Therefore, 3/4 × 2/5 = 6/20 = 3/10.

Worked example 4. A tortoise covers 2/5 km in one hour. Find its distance in 3/4 hour at the same pace.

Answer: In 1/4 hour it covers 2/20 km, found by dividing 2/5 into four equal parts. In 3/4 hour it covers 3 × 2/20 = 6/20 = 3/10 km.

The two stages have different purposes. Dividing by the multiplier's denominator selects one equal share. Multiplying by its numerator selects the required number of shares. The square shows why both stages are needed.

This interpretation also works for fractions greater than one. Dividing 3/2 into four equal parts gives 3/8. Taking five such parts gives 5/4 × 3/2 = 15/8.

What is the general rule for multiplying fractions?

Result: Multiply the numerators and multiply the denominators

Let a and c be the numerators of two fractions, and b and d their respective denominators. The letters represent whole numbers, with b and d not zero. The general multiplication rule is a/b × c/d = (a × c)/(b × d).

The numerator of the product counts the selected small parts. Its denominator counts how many equal small parts make one whole. The area model explains why multiplying the two denominators gives that total number of parts.

For two unit fractions, both numerators are 1. Consequently, their product is a unit fraction whose denominator is the product of their denominators: 1/b × 1/d = 1/(b × d).

Worked example 5. Find 1/12 × 1/18.

Answer: A whole divided into 18 rows and 12 columns has 18 × 12 = 216 equal parts. Selecting one of them gives 1/216. Thus, 1/12 × 1/18 = 1/216.

How do we include whole numbers?

A whole number can be written as a fraction with denominator 1. This brings multiplication involving whole numbers under the same rule. There is no need for a separate multiplication formula.

In each row, the whole-number factor becomes a numerator over 1. Multiplying by this denominator leaves the other denominator unchanged, which agrees with the earlier method of adding equal fractional quantities.

Brahmagupta gave the general multiplication formula in the Brāhmasphuṭasiddhānta in 628 CE. CE means Common Era, the calendar designation used with this date. The rule applies to two or more fractions by multiplying their numerators together and their denominators together.

How can common factors make multiplication easier?

A factor of a whole number divides it exactly. A common factor divides two numbers exactly. Cancelling common factors means dividing a numerator and a denominator in a product by the same common factor before completing the multiplication.

Equivalent fractions have the same value although their numerators and denominators differ. Dividing a fraction's numerator and denominator by the same non-zero common factor preserves its value. This is the reason cancellation works.

How do we simplify to lowest form?

A fraction is in its lowest form when its numerator and denominator have no common factor greater than 1. Simplifying before multiplication can keep the numbers in the calculation smaller while giving the same final answer.

Worked example 6. Multiply 12/7 × 5/24 and express the product in lowest form.

Answer: Write the product as (12 × 5)/(7 × 24). Divide 12 and 24 by their common factor 12, leaving (1 × 5)/(7 × 2). The product is 5/14.

The cancellation does not discard part of the fraction. It divides the combined numerator and denominator by the same number. The simplified multiplication therefore represents exactly the same quantity as the original multiplication.

Worked example 7. Simplify 14/15 × 25/42.

Answer: Write (14 × 25)/(15 × 42). Divide 14 and 42 by 14, giving 1 and 3. Divide 25 and 15 by 5, giving 5 and 3. The result is (1 × 5)/(3 × 3) = 5/9.

Here cancellation uses numbers from different original fractions. This is valid because they are factors of the combined numerator and denominator. Finish by multiplying the remaining numerators and the remaining denominators.

Note: Cancellation requires a common factor in a numerator and a denominator. Dividing two numerators alone by a common factor changes the value of the product.

When is a product greater or smaller than its factors?

A factor in a multiplication is a number being multiplied. For the positive numbers considered here, compare each factor with 1 before estimating the product. Positive means greater than zero.

The symbol < means “less than”, and > means “greater than”. Multiplication need not produce an answer greater than both factors. A multiplier between 0 and 1 takes only part of the other factor.

Property: The size of the multiplier determines the change

Multiplying a positive quantity by a number between 0 and 1 gives a smaller quantity. Multiplying it by a number greater than 1 gives a larger quantity. Multiplication by 1 leaves the quantity unchanged.

For 3/4 × 2/5, the product is 6/20. Writing the factors with the same denominator gives 3/4 = 15/20 and 2/5 = 8/20. Six twentieths is smaller than either fifteen twentieths or eight twentieths.

By contrast, 1/4 × 8 = 2 lies between its factors. The product exceeds 1/4 but is below 8. Comparing with each factor separately prevents the misleading statement that a product is simply “bigger” or “smaller” without saying than what.

How are rectangular area and the order of multiplication connected?

The length and breadth of a rectangle are its two side measurements. Its area is their product. The same rule applies when either side, or both sides, are fractional lengths.

A square is divided into four rows and two columns. The top row is yellow, and its left half is hatched. The hatched rectangle is labelled 1/8.

See Fig. 8.3 in your NCERT textbook

The hatched rectangle measures 1/2 unit by 1/4 unit. Eight such rectangles fill the unit square. Its area is therefore 1/2 × 1/4 = 1/8 square unit. “Square unit” expresses area rather than side length.

Property: Interchanging factors leaves their product unchanged

Interchanging a rectangle's length and breadth leaves its area unchanged. Likewise, a/b × c/d = c/d × a/b, using the numerator and denominator letters defined earlier. Both multiplication orders describe the same area.

This also follows from multiplying the numerators together and denominators together. Reversing the order of those whole-number multiplications does not change their products. Thus 1/2 × 1/4 and 1/4 × 1/2 both give 1/8.

Worked example 8. Find the area of a rectangle with sides 3 3/4 ft and 9 3/5 ft. The abbreviation ft means feet, a unit of length.

Answer: Convert the side lengths to 15/4 ft and 48/5 ft. Multiply: (15 × 48)/(4 × 5) = 3 × 12 = 36. The area is 36 square feet.

Keep the meaning of the answer in view: multiplication of two side lengths gives an area. It does not give another side length. Converting both mixed fractions first makes every part of each side enter the calculation.

How does a reciprocal turn division into multiplication?

In division, the dividend is the number being divided, the divisor is the number we divide by, and the quotient is the result. In 12 ÷ 4 = 3, these numbers are 12, 4 and 3 respectively.

Division can be understood as finding a missing factor. The question 12 ÷ 4 asks what number multiplied by 4 gives 12. The same approach helps when a divisor is a fraction.

What number makes the product equal to one?

Definition: The reciprocal of a non-zero fraction is obtained by interchanging its numerator and denominator. A fraction multiplied by its reciprocal gives 1.

For instance, the reciprocal of 2/3 is 3/2 because 2/3 × 3/2 = 1. Therefore, 1 ÷ 2/3 = 3/2. To make the product 3 instead of 1, multiply this reciprocal by 3.

Worked example 9. Evaluate 3 ÷ 2/3 by finding a missing factor.

Answer: We need a number which, when multiplied by 2/3, gives 3. Multiplying 2/3 by 3/2 gives 1. Multiplying by 3/2 × 3 therefore gives 3. The quotient is 9/2.

Result: Multiply the dividend by the divisor's reciprocal

With the earlier letters, a/b ÷ c/d = a/b × d/c = (a × d)/(b × c). In this division formula, c must also be non-zero because the divisor must not be zero.

1. Identify the dividend and divisor in their given order.
2. Find the reciprocal of the divisor by interchanging its numerator and denominator.
3. Multiply the unchanged dividend by that reciprocal.
4. Cancel common factors where possible and simplify the product.

For example, 2/3 ÷ 3/5 = 2/3 × 5/3 = 10/9. The reciprocal is taken of 3/5, because that is the divisor. The dividend remains 2/3 throughout.

When does division increase or decrease a quantity?

For a positive dividend, the divisor's relation to 1 determines whether the quotient is greater or smaller than the dividend. Dividing by a fraction between 0 and 1 means multiplying by a reciprocal greater than 1.

Property: Compare the divisor with one

If the divisor is between 0 and 1, the quotient is greater than the positive dividend. If the divisor is greater than 1, the quotient is less than the positive dividend. This follows directly from the multiplication comparison rules.

In the last row, both the dividend and divisor are below one, yet the answer exceeds the dividend. It is the divisor's size relative to 1 that explains this increase, not whether the dividend itself is a whole number.

These comparisons are useful checks before and after calculation. For 1/5 ÷ 1/2, the divisor is between 0 and 1, so the quotient should exceed 1/5. Multiplying by the reciprocal gives 2/5, which satisfies that check.

Note: Keep multiplication and division comparisons separate. Multiplying by a positive fraction below one makes a positive quantity smaller; dividing by that fraction makes it larger.

How do we choose the operation in a sharing problem?

Read what the answer represents before choosing an operation. Equal sharing asks for the amount in one share when the total and number of shares are known. Another type of division asks how many fixed-size shares fit into a total.

How much is in one share?

Worked example 10. Leena uses 1/4 litre of milk in five cups of tea, with equal amounts of milk in the cups. A litre is a unit of volume, the amount of space a liquid occupies. Find the milk in each cup.

Answer: Divide the total milk by five. Since the reciprocal of 5 is 1/5, the calculation is 1/4 ÷ 5 = 1/4 × 1/5 = 1/20. Each cup contains 1/20 litre of milk.

The divisor counts cups, while the dividend measures milk. The quotient is a volume per cup. Multiplying that share by five returns the total volume, providing a direct check of the interpretation.

How many equal shares can be made?

Maria uses 1/4 metre of lace for each bag, with 8 metres available. A metre is a unit of length, abbreviated m. The number of bags is 8 ÷ 1/4 = 32, because the calculation counts quarter-metre pieces.

A baker with 5 kilograms of flour, using 1/6 kilogram per loaf, makes 5 ÷ 1/6 = 30 loaves. A kilogram, abbreviated kg, is a unit of mass. The quotient here counts loaves rather than measuring flour.

Mariam and her cousins eat 4/5 of a cake. Its remaining fraction is 1 − 4/5 = 1/5. Sharing this remainder equally among three friends gives 1/5 ÷ 3 = 1/15 of the original cake per friend.

How many bricks cover a given area?

Worked example 11. Cover an area of 7 1/2 square units with square bricks whose sides are each 1/5 unit. Find the number of bricks represented by this area calculation.

Answer: Each brick has area 1/5 × 1/5 = 1/25 square unit. The total area is 15/2 square units. Divide total area by brick area: 15/2 ÷ 1/25 = 15/2 × 25 = 375/2 bricks.

This calculation combines multiplication and division. Multiplication finds the area of a single brick from its side lengths. Division then compares the total area with that single-brick area. Dividing by a side length instead would compare unlike measurements.

The relation is: number of bricks multiplied by area of one brick equals total area. Keeping this relation in mind explains why the total area is the dividend and the area of one brick is the divisor.

How do fractions describe several fountains working together?

A cistern is a tank that holds water. To combine fountains filling it, express each fountain's contribution over the same time interval. A filling rate tells how much of the cistern is filled in a given time.

Four fountains can individually fill a cistern in one day, half a day, a quarter of a day and one fifth of a day. Their filling times differ, so first convert each time into the number of cisterns it could fill in one day.

Why do we add rates rather than filling times?

A fountain taking half a day can fill the cistern twice in one day. A fountain taking a quarter of a day can fill it four times. These quantities measure contributions over the same interval and can be added.

Worked example 12. Four fountains fill one cistern individually in 1 day, 1/2 day, 1/4 day and 1/5 day. With their filling rates maintained, find the time taken when all flow together.

Answer: Their daily filling capacities are 1 ÷ 1 = 1, 1 ÷ 1/2 = 2, 1 ÷ 1/4 = 4 and 1 ÷ 1/5 = 5 cisterns. Together they fill 1 + 2 + 4 + 5 = 12 cisterns per day. One cistern therefore takes 1/12 day.

The first use of division converts each individual time into a rate. Addition combines those rates. The final division converts the combined rate back into the time for one cistern. Each operation answers a different question.

Adding the four given times would describe successive time intervals. It would not describe fountains flowing together. The common one-day interval is what makes the four contributions comparable.

How can successive fractions identify a shaded area?

A shaded region can be a fraction of a smaller shape which is itself a fraction of the whole. Identify each intermediate whole carefully. Multiplication then converts these successive fractional relationships into a fraction of the original area.

Lines divide a large square, with hatching in its upper-right part. A red outline marks the top-right smaller square. An enlarged view colours a triangle inside this smaller square yellow.

See Figs. 8.4, 8.5 and 8.6 in your NCERT textbook

Which whole does each fraction refer to?

Take the large square's area as 1 square unit. The red square occupies 1/4 of that area. The yellow triangle occupies half of the red square, so its area is 1/2 × 1/4 = 1/8 square unit.

The hatched region occupies 3/4 of the yellow triangle. Its area is therefore 3/4 × 1/8 = 3/32 square unit. Thus it occupies 3/32 of the large square.

It would be incorrect to call the hatching three quarters of the large square. The fraction 3/4 describes its relationship to the yellow triangle. Multiplying by 1/8 accounts for the triangle's relationship to the original whole.

This chain of reasoning uses the same idea as taking a fraction of a quantity: first identify the quantity being divided, then select the required part. Keeping the original whole visible prevents the fractions from being attached to the wrong region.

How do successive fractions work in the dramma problem?

A dramma was a silver coin. In a donation problem from Bhāskara II's Līlāvatī, written in 1150 CE, a miser gives successive fractional parts of a dramma. A cowrie shell is the small-value item used to express the final gift.

The tale uses the relation 1 dramma = 1280 cowrie shells. Treat this as the conversion in this particular problem. Historical conversion rates between coins varied with region, time period, economic conditions, coin weights and purity.

How do we calculate a fraction of a fraction?

Worked example 13. A gift is 1/5 of 1/16 of 1/4 of 1/2 of 2/3 of 3/4 of a dramma. In this tale, one dramma equals 1280 cowrie shells. Find the gift in cowrie shells.

Answer: Replace each “of” by multiplication. The fractional gift is 1/5 × 1/16 × 1/4 × 1/2 × 2/3 × 3/4 = 6/7680 = 1/1280 of a dramma. This is one cowrie shell.

Changing the order of the multiplications does not change their product. This allows common factors to be cancelled before carrying out all the multiplication. It also connects a long chain of fractional parts with the same rule used for two fractions.

The small final amount explains the humour in the donation story: the long description of the gift reduces to one cowrie shell. The calculation depends on the stated conversion, so the conversion must accompany the problem whenever it is solved.

For any problem involving successive fractional parts, keep track of the starting whole. Each multiplication narrows or scales the current quantity. Convert the final fraction into a count or measurement only after establishing the relation to that original whole.

Glossary

• Fraction — A number expressed using a numerator and a non-zero denominator to describe equal parts of a whole.
• Numerator — The number above the fraction bar, counting the equal fractional parts being considered.
• Denominator — The number below the fraction bar, indicating how many equal parts make one whole.
• Unit fraction — A fraction with numerator one, representing one equal part of a whole.
• Mixed fraction — A number written using a whole-number part together with a fractional part.
• Product — The result obtained when two or more numbers are multiplied together.
• Multiplicand — The quantity whose multiple or fractional part is taken in a multiplication.
• Multiplier — The number specifying which multiple or fractional part of the multiplicand to take.
• Common factor — A number that divides each of two whole numbers exactly, leaving no remainder.
• Lowest form — A fraction's form in which numerator and denominator share no common factor greater than one.
• Reciprocal — The number obtained by interchanging a non-zero fraction's numerator and denominator, giving product one with the original fraction.
• Dividend — The quantity being divided in a division, before the divisor is applied.
• Divisor — The non-zero quantity by which the dividend is divided to obtain the quotient.
• Quotient — The result obtained by dividing the dividend by the given non-zero divisor.

Common errors and misconceptions

• Misconception: Multiplication makes a number greater. Correct: Multiplying a positive quantity by a fraction between zero and one makes it smaller. For example, 1/4 × 8 = 2.
• Misconception: Division makes a number smaller. Correct: Dividing a positive quantity by a fraction between zero and one increases it, as 6 ÷ 1/4 = 24 shows.
• Misconception: To divide fractions, take the reciprocal of the dividend. Correct: Keep the dividend unchanged and multiply it by the divisor's reciprocal: 2/3 ÷ 3/5 = 2/3 × 5/3.
• Misconception: A mixed fraction can be multiplied by using only its fractional part. Correct: Include the whole-number part too; 1 1/4 becomes 5/4 before multiplication.
• Misconception: Any two numbers in a product of fractions can be cancelled. Correct: Cancel common factors between a numerator and a denominator to preserve the product's value.
• Misconception: A fraction of a smaller region is the same fraction of the whole diagram. Correct: Multiply by the smaller region's fraction of the whole, as 3/4 × 1/8 = 3/32 illustrates.
• Misconception: Add individual filling times to find the time when fountains work together. Correct: First add their contributions over the same time interval, then find the time for one cistern.

Exam-style questions with model answers

Q1. A tortoise travels 1/4 km in one hour. At the same pace, how far does it travel in three hours? Show the repeated addition and the product. [2 marks]

1. The three equal hourly distances add to 1/4 + 1/4 + 1/4 = 3/4 km.
2. This is the multiplication 3 × 1/4 = 3/4. Therefore, the tortoise travels 3/4 km in three hours.

Q2. Internet time costs ₹8 per hour at a fixed rate. Find the cost of 1 1/4 hours, showing the mixed-fraction conversion and multiplication. [3 marks]

1. Convert the time into a single fraction: 1 1/4 = 5/4 hours, since the whole hour contains four quarter-hours.
2. Multiply the time by the hourly cost: 5/4 × 8 = (5 × 8)/4.
3. Divide 8 by 4 to get 2, then multiply by 5 to get 10. The required cost is ₹10.

Q3. A unit square has five equal rows and four equal columns. Use it to explain 3/4 × 2/5, then compare the product with both factors. [4 marks]

1. The square contains 5 × 4 = 20 equal small rectangles. Two of the five rows represent 2/5 of the whole.
2. One quarter of these two rows contains two small rectangles. Three quarters therefore contains six, giving the product 6/20.
3. Simplifying gives 6/20 = 3/10. This is the fraction of the whole represented by 3/4 × 2/5.
4. The factors are 3/4 = 15/20 and 2/5 = 8/20. Since 6 is smaller than both 15 and 8, the product is less than both factors.

Q4. Calculate 14/15 × 25/42 in lowest form by cancelling common factors. Explain why this cancellation preserves the value. [3 marks]

1. Write the product as (14 × 25)/(15 × 42). Divide the numerator factor 14 and denominator factor 42 by 14, leaving 1 and 3.
2. Divide the numerator factor 25 and denominator factor 15 by 5, leaving 5 and 3. The remaining product is (1 × 5)/(3 × 3) = 5/9.
3. Each cancellation divides the combined numerator and denominator by the same non-zero factor. The fraction's value is therefore unchanged, and 5/9 is in lowest form.

Q5. Evaluate 2/3 ÷ 3/5. Identify the dividend and divisor, explain the reciprocal step, and compare the quotient with the dividend. [4 marks]

1. The dividend is 2/3 and the divisor is 3/5. Their order determines which fraction must be replaced by its reciprocal.
2. The reciprocal of 3/5 is 5/3, since 3/5 × 5/3 = 1. The dividend remains unchanged.
3. Multiply the dividend by this reciprocal: 2/3 × 5/3 = (2 × 5)/(3 × 3) = 10/9.
4. The quotient is greater than the dividend because the divisor lies between zero and one. Indeed, 2/3 = 6/9, and 10/9 is greater than 6/9.

Q6. The government takes 1/6 of Somu's original land for a road. Of the land remaining, she gives 1/2 to Krishna and 1/3 to Bora, keeping the rest. Find each person's fraction of the original land and check the total, including the road. [5 marks]

1. Take the original land as one whole. After the road takes 1/6, the fraction remaining for Somu to distribute is 1 − 1/6 = 5/6.
2. Krishna receives half of this remainder, not half of the original land. Her share is 1/2 × 5/6 = 5/12 of the original land.
3. Bora receives one third of the same remainder. His share is 1/3 × 5/6 = 5/18 of the original land.
4. The fraction of the remainder kept by Somu is 1 − 1/2 − 1/3 = 1/6. Her share of the original land is 1/6 × 5/6 = 5/36.
5. Using denominator 36 checks every share: road 6/36, Krishna 15/36, Bora 10/36 and Somu 5/36. Their sum is 36/36 = 1.

Q7. Four fountains individually fill a cistern in 1 day, 1/2 day, 1/4 day and 1/5 day. Starting with an empty cistern, all four flow together at their unchanged rates, with no water leaving it. Calculate each daily contribution and the combined filling time. [5 marks]

1. The first fountain fills one cistern in one day. Its daily contribution is therefore 1 ÷ 1 = 1 cistern.
2. The second fountain takes half a day for one cistern. In one day it can fill 1 ÷ 1/2 = 2 cisterns.
3. The third fountain takes a quarter of a day. Its daily contribution is 1 ÷ 1/4 = 4 cisterns.
4. The fourth fountain takes one fifth of a day. Its daily contribution is 1 ÷ 1/5 = 5 cisterns.
5. The combined daily contribution is 1 + 2 + 4 + 5 = 12 cisterns. The time for one cistern is therefore 1 ÷ 12 = 1/12 day.

Q8. A donation is 1/5 of 1/16 of 1/4 of 1/2 of 2/3 of 3/4 of a dramma. Given that one dramma equals 1280 cowrie shells in this problem, find the donation in cowrie shells. [3 marks]

1. Each “of” means multiplication. The gift as a fraction of a dramma is 1/5 × 1/16 × 1/4 × 1/2 × 2/3 × 3/4.
2. The product of the numerators is 6 and that of the denominators is 7680. The resulting fraction 6/7680 simplifies to 1/1280.
3. Using the given conversion, 1/1280 of 1280 cowrie shells is one cowrie shell. This is the complete donation.

Key takeaways

• Multiplying by a fraction means dividing the quantity into equal shares and taking the required number of those shares.
• A unit-square model shows why the denominators multiply to count equal small parts and the numerators multiply to count selected parts.
• Write whole numbers over one and convert mixed fractions into single fractions before using the general multiplication rule.
• Cancelling common factors between a numerator and a denominator preserves the product and can make multiplication easier.
• The order of multiplication does not affect its result, just as interchanging a rectangle's side measurements does not change its area.
• Divide by multiplying the unchanged dividend by the divisor's reciprocal, then simplify the resulting product.
• For positive quantities, multiplication by a fraction between zero and one decreases the quantity, while division by it increases the quantity.
• In sharing, filling and shaded-area problems, identify the relevant whole and state what the final number measures or counts.

Test yourself

Why does 1/2 × 1/4 equal 1/8?

A quarter divided into two equal parts gives eighths of the original whole. Taking one of those parts gives one eighth.

What is 5/4 × 3/2?

The product is 15/8. Multiply the numerators to get 15 and the denominators to get 8.

Which common factor helps simplify 12/7 × 5/24 before multiplication?

Cancel the common factor 12 between 12 and 24. The remaining calculation is (1 × 5)/(7 × 2) = 5/14.

What is the reciprocal of 3/5, and why?

The reciprocal is 5/3 because multiplying 3/5 by 5/3 gives a product of one.

Does 6 ÷ 1/4 give a number greater or smaller than 6?

It gives 24, which is greater than 6. Division by one quarter is multiplication by four.

Maria has 8 metres of lace and uses 1/4 metre per bag. How many bags can she decorate?

She can decorate 32 bags, since 8 ÷ 1/4 = 8 × 4 = 32.

Mariam and her cousins eat 4/5 of a cake. Three friends share the remainder equally. What fraction of the original cake does each friend receive?

Each receives 1/15 of the cake. The remainder is 1/5, and 1/5 divided equally among three gives 1/15.

A yellow triangle occupies 1/8 of a square, and 3/4 of that triangle is shaded. What fraction of the square is shaded?

The shaded fraction is 3/4 × 1/8 = 3/32 of the original square.