Unitary Method

12 bags of rice having equal weights weigh 174 kg. How much will 20 such bags weigh?

Answer

Weight of 12 bags = 174 kg.

Weight of 1 bag = $\Big(\dfrac{174}{12}\Big) \text{ kg} \quad \text{[Less bags, Less weight]}$

Weight of 20 bags = $\Big(\dfrac{174}{12} \times 20\Big) \text{ kg} \quad \text{[More bags, More weight]}$

= $\Big(\dfrac{174}{3} \times 5\Big)$ kg

= 58 x 5 kg $\quad$ (Dividing 174 and 3 by 3)

= 290 kg.

Hence, the weight of 20 bags is 290 kg.

8 kg of tomatoes cost ₹ 84. What is the cost of 14 kg of tomatoes?

Answer

Cost of 8 kg tomatoes = ₹ 84.

Cost of 1 kg tomato = ₹ $\Big(\dfrac{84}{8}\Big) \quad \text{[Less tomatoes, Less cost]}$

Cost of 14 kg tomatoes = ₹ $\Big(\dfrac{84}{8} \times 14\Big) \quad \text{[More tomatoes, More cost]}$

= ₹ $\Big(\dfrac{21}{2} \times 14\Big)$

= ₹ $\Big(\dfrac{294}{2}\Big)$

= ₹ 147

Hence, the cost of 14 kg of tomatoes is ₹ 147.

16 m of cloth costs ₹ 2200. What is the cost of 6 m of cloth?

Answer

Cost of 16 m of cloth = ₹ 2200.

Cost of 1 m of cloth = ₹ $\Big(\dfrac{2200}{16}\Big) \quad \text{[Less cloth, Less cost]}$

Cost of 6 m of cloth = ₹ $\Big(\dfrac{2200}{16} \times 6\Big) \quad \text{[More cloth, More cost]}$

$= ₹ \Big(\dfrac{2200}{16} \times 6\Big) \\[1em] = ₹ \Big(\dfrac{275}{2} \times 6\Big) \quad \text{(Dividing 2200 and 16 by 8)} \\[1em] = ₹ \Big(\dfrac{1650}{2}\Big) \\[1em] = ₹ 825$

Hence, the cost of 6 m of cloth is ₹ 825.

If 8.5 m of a uniform rod weighs 30.6 kg, what will be the weight of 5 m of the same rod?

Answer

Weight of 8.5 m rod = 30.6 kg.

Weight of 1 m rod = $\Big(\dfrac{30.6}{8.5}\Big) \text{ kg} \quad \text{[Less length, Less weight]}$

Multiply by 10 to remove decimals.

Weight of 1 m rod = $\Big(\dfrac{30.6 \times 10}{8.5 \times 10}\Big) \text{ kg} = \Big(\dfrac{306}{85}\Big) \text{ kg}$

Weight of 5 m rod = $\Big(\dfrac{306}{85} \times 5\Big) \text{ kg} \quad \text{[More length, More weight]}$

= $\Big(\dfrac{306}{17}\Big) \text{ kg} \quad \text{(Dividing 5 and 85 by 5)}$

= 18 kg.

Hence, the weight of 5 m of the rod is 18 kg.

The cost of 6 purses is ₹ 453. How many such purses can be bought for ₹ 604?

Answer

For ₹ 453, number of purses bought = 6.

For ₹ 1, number of purses bought = $\Big(\dfrac{6}{453}\Big) \text{ purses} \quad \text{[Less money, Less purses]}$

For ₹ 604, number of purses bought = $\Big(\dfrac{6}{453} \times 604\Big) \text{ purses} \quad \text{[More money, More purses]}$

= $\Big(\dfrac{2}{151} \times 604\Big) \text{ purses} \quad \text{(Dividing 6 and 453 by 3)}$

= $\Big(\dfrac{1208}{151}\Big)$ purses

= 8 purses.

Hence, 8 purses can be bought for ₹ 604.

12 apples weigh 2 kg 500 g. How many apples will weigh 15 kg?

Answer

First, convert weight to a single unit:

1000 g = 1 kg

500 g = 0.5 kg

∴ 2 kg 500 g = (2 + 0.5) kg = 2.5 kg.

For 2.5 kg, number of apples = 12.

For 1 kg, number of apples = $\Big(\dfrac{12}{2.5}\Big)$. $\quad$[Less weight, Less apples]

Multiply by 10 to remove decimals.

For 1 kg, number of apples = $\Big(\dfrac{12 \times 10}{2.5 \times 10}\Big) = \Big(\dfrac{120}{25}\Big)$ apples

For 15 kg, number of apples = $\Big(\dfrac{120}{25} \times 15\Big) \text{ apples} \quad \text{[More weight, More apples]}$

= $\Big(\dfrac{120}{5} \times 3\Big) \text{ apples} \quad \text{(Dividing 15 and 25 by 5)}$

= 24 x 3 apples $\quad$ (Dividing 120 and 5 by 5)

= 72 apples.

Hence, 72 apples will weigh 15 kg.

55 m of cloth is required to make 25 shirts. How much cloth will be required to make 35 shirts of the same size?

Answer

Cloth required for 25 shirts = 55 m.

Cloth required for 1 shirt = $\Big( \dfrac{55}{25}\Big) \text{m} \quad \text{[Less shirts, Less cloth]}$

Cloth required for 35 shirts = $\Big( \dfrac{55}{25} \times 35\Big) \text{m} \quad \text{[More shirts, More cloth]}$

= $\Big( \dfrac{55}{5} \times 7\Big)$ m

= 11 x 7 m $\quad$ (Dividing 55 and 5 by 5)

= 77 m.

Hence, 77 m of cloth will be required to make 35 shirts.

A scooter consumes 2.5 litres of petrol in covering a distance of 85 km. What distance will it cover in 8 litres of petrol?

Answer

In 2.5 litres, distance covered = 85 km.

In 1 litre, distance covered = $\Big( \dfrac{85}{2.5}\Big)$ km. $\quad$[Less petrol, Less distance]

Multiply by 10 to remove decimals.

In 1 litre, distance covered = $\Big( \dfrac{85 \times 10}{2.5 \times 10}\Big) = \Big( \dfrac{850}{25}\Big)$ km

In 8 litres, distance covered = $\Big( \dfrac{850}{25} \times 8\Big) \text{ km} \quad \text{[More petrol, More distance]}$

= 34 x 8 km $\quad$ (Dividing 850 and 25 by 25)

= 272 km.

Hence, the scooter will cover 272 km in 8 litres of petrol.

A journey of 124 km costs ₹ 1395. How much will a journey of 240 km cost?

Answer

Cost of 124 km journey = ₹ 1395.

Cost of 1 km journey = ₹ $\Big(\dfrac{1395}{124}\Big) \quad \text{[Less distance, Less cost]}$

Cost of 240 km journey = ₹ $\Big(\dfrac{1395}{124} \times 240\Big) \quad \text{[More distance, More cost]}$

= ₹ 11.25 x 240 $\quad$ (Dividing 1395 and 124 by 124)

= ₹ 2700

Hence, a journey of 240 km will cost ₹ 2700.

A compositor takes 1 hour 45 minutes to compose 24 pages of a book. How long will he take to compose 64 pages?

Answer

First, convert time into minutes:

1 hour = 60 minutes

∴ 1 hour 45 minutes = 60 + 45 = 105 minutes.

Time taken to compose 24 pages = 105 minutes.

Time taken to compose 1 page = $\Big( \dfrac{105}{24}\Big) \text{ minutes} \quad \text{[Less pages, Less time]}$

Time taken to compose 64 pages = $\Big(\dfrac{105}{24} \times 64\Big) \text{ minutes} \quad \text{[More pages, More time]}$

= $\Big(\dfrac{105}{3} \times 8\Big) \text{ minutes} \quad \text{(Dividing 24 and 64 by 8)}$

= 35 x 8 minutes

= 280 minutes (or 4 hours 40 minutes)

Hence, he will take 4 hours 40 minutes to compose 64 pages.

If a man walks 16 km in 5 hours, how long would he take to walk 11.2 km?

Answer

Time taken to walk 16 km = 5 hours.

Time taken to walk 1 km = $\Big(\dfrac{5}{16}\Big) \text{ hours} \quad \text{[Less distance, Less time]}$

Time taken to walk 11.2 km = $\Big(\dfrac{5}{16} \times 11.2\Big) \text{ hours} \quad \text{[More distance, More time]}$

= $\Big(\dfrac{5 \times 11.2}{16}\Big)$ hours

= $\dfrac{56}{16}$ hours

= 3.5 hours (or 3 hours 30 minutes).

Time taken to walk 11.2 km = 3 hours 30 minutes

5 men can paint a hall in 18 hours. How many men will be able to paint it in 10 hours?

Answer

Men required to paint in 18 hours = 5 men.

Men required to paint in 1 hour = (5 x 18) men. $\quad$ [Less time, More men]

Men required to paint in 10 hours = $\Big(\dfrac{5 \times 18}{10}\Big) \text{ men} \quad \text{[More time, Less men]}$

= $\dfrac{90}{10}$ men

= 9 men.

9 men are required to paint in 10 hours.

30 workers can dig a trench in 5 days. How many workers will be required to dig it in 6 days?

Answer

Workers required to dig in 5 days = 30 workers.

Workers required to dig in 1 day = (30 x 5) workers. $\quad$ [Less days, More workers]

Workers required to dig in 6 days = $\Big(\dfrac{30 \times 5}{6}\Big) \text{ workers} \quad \text{[More days, Less workers]}$

= (5 x 5) workers $\quad$ [Dividing 30 and 6 by 6]

= 25 workers.

25 workers are required to dig in 6 days.

In a fort, 360 men had provisions for 21 days. If 60 more men join them, how long will the provisions last?

Answer

Original men = 360.

New men = 360 + 60 = 420.

Provisions for 360 men last = 21 days.

Provisions for 1 man last = (21 x 360) days. $\quad$ [Less men, More days]

Provisions for 420 men last = $\Big(\dfrac{21 \times 360}{420}\Big) \text{ days} \quad \text{[More men, Less days]}$

= $\Big(\dfrac{1 \times 360}{20}\Big) \text{ days} \quad \text{[Dividing 21 and 420 by 21]}$

= 18 days.

420 men had provisions for 18 days.

In an army camp, 300 soldiers had provisions for 13 days. If 40 of them are transferred to the other camp, how long will the provisions last?

Answer

Original soldiers = 300.

Remaining soldiers = 300 - 40 = 260.

Provisions for 300 soldiers last = 13 days.

Provisions for 1 soldier last = (13 x 300) days. $\quad$ [Less soldiers, More days]

Provisions for 260 soldiers last = $\Big(\dfrac{13 \times 300}{260}\Big) \text{ days} \quad \text{[More soldiers, Less days]}$

= $\Big(\dfrac{1 \times 300}{20}\Big) \text{ days} \quad \text{[Dividing 13 and 260 by 13]}$

= 15 days.

260 soldiers had provisions for 15 days

Moving at the rate of 70 km/hr, a car completes a journey in 18 minutes. How long would it take to complete this journey, if the speed is increased to 84 km/hr?

Answer

Time taken at 70 km/hr speed = 18 minutes.

Time taken at 1 km/hr speed = (18 x 70) minutes. $\quad$ [Less speed, More time]

Time taken at 84 km/hr speed = $\Big(\dfrac{18 \times 70}{84}\Big) \text{ minutes} \quad \text{[More speed, Less time]}$

= $\Big(\dfrac{3 \times 70}{14}\Big) \text{ minutes} \quad \text{[Dividing 18 and 84 by 6]}$

= 3 x 5 minutes $\quad$ [Dividing 70 and 14 by 14]

= 15 minutes.

At the rate of 84 km/hr, the car completes the journey in 15 minutes.

If 15 men can level a ground in 60 days, in how many days can 36 men level the same ground?

1. 24 days
2. 25 days
3. 27 days
4. 30 days

Answer

Given:

Days taken by 15 men = 60 days

Days taken by 1 man = 60 x 15 days $\quad$[Less men, More days]

Days taken by 36 men = $\Big(\dfrac{60 \times 15}{36}\Big) \text{ days} \quad \text{[More men, Less days]}$

= $\Big(\dfrac{5 \times 15}{3}\Big) \text{ days} \quad \text{[Dividing 60 and 36 by 12]}$

= 5 x 5 days $\quad$ [Dividing 15 and 3 by 3]

= 25 days

Hence, option 2 is the correct option.

If 150 m of cloth is required to prepare dresses for 42 women, then for how many women will 125 m of cloth be sufficient?

1. 30
2. 32
3. 35
4. 36

Answer

Given:

Number of women for 150 m cloth = 42 women

Number of women for 1 m cloth = $\dfrac{42}{150} \text{ women} \quad \text{[Less cloth, Less women]}$

Number of women for 125 m cloth = $\Big(\dfrac{42}{150} \times 125\Big) \text{ women} \quad \text{[More cloth, More women]}$

= $\Big(\dfrac{42}{6} \times 5\Big) \text{ women} \quad \text{[Dividing 125 and 150 by 25]}$

= 7 x 5 women $\quad$ [Dividing 42 and 6 by 6]

= 35 women

Hence, option 3 is the correct option.

In a map, 1.5 cm represents 46.8 km. How much distance will be represented by 3.5 cm on the map?

1. 96.4 km
2. 98.5 km
3. 109.2 km
4. 113.6 km

Answer

Given:

Distance for 1.5 cm = 46.8 km

Distance for 1 cm = $\dfrac{46.8}{1.5} \text{ km} \quad \text{[Less cm, Less distance]}$

Distance for 3.5 cm = $\Big(\dfrac{46.8}{1.5} \times 3.5\Big) \text{ km} \quad \text{[More cm, More distance]}$

Multiply by 10 to remove decimals:

= $\Big(\dfrac{46.8 \times 10}{1.5 \times 10} \times 3.5\Big) \text{ km} = \Big(\dfrac{468}{15} \times 3.5\Big)$ km

= 31.2 x 3.5 km $\quad$ [Dividing 468 and 15 by 15]

= 109.2 km

Hence, option 3 is the correct option.

If a car can go 224 km in 20 litres of petrol, how far can it go in 32.5 litres of petrol?

1. 364 km
2. 414 km
3. 288 km
4. 298 km

Answer

Distance on 20 L = 224 km

Distance on 1 L = $\dfrac{224}{20} \text{ km} \quad \text{[Less petrol, Less distance]}$

Distance on 32.5 L = $\Big(\dfrac{224}{20} \times 32.5\Big) \text{ km} \quad \text{[More petrol, More distance]}$

= 11.2 x 32.5 km $\quad$ [Dividing 224 and 20 by 20]

= 364 km

Hence, option 1 is the correct option.

If 16 buffaloes eat as much as 28 cows, how many buffaloes eat as much as 91 cows?

1. 48
2. 52
3. 64
4. 76

Answer

Buffaloes for 28 cows = 16 Buffaloes

Buffaloes for 1 cow = $\dfrac{16}{28} \text{ Buffaloes} \quad \text{[Less cows, Less buffaloes]}$

Buffaloes for 91 cows = $\Big(\dfrac{16}{28} \times 91\Big) \text{ Buffaloes} \quad \text{[More cows, More buffaloes]}$

= $\dfrac{4}{7} \times 91 \text{ Buffaloes} \quad \text{[Dividing 16 and 28 by 4]}$

= 4 x 13 Buffaloes $\quad$ [Dividing 91 and 7 by 7]

= 52 Buffaloes

Hence, option 2 is the correct option.

Fill in the blanks :

(i) If 6 pens cost ₹ 69, then the cost of 16 pens is ............... .

(ii) If 1 dozen eggs cost ₹ 54, then a tray of 30 eggs will cost ............... .

(iii) A worker is paid ₹ 1610 as wages for 14 days. His wages for 30 days will be ............... .

(iv) 25 boxes of 12 ice-cream cups each, cost ₹ 10500. The cost of 15 boxes of 20 ice-cream cups each, will be ............... .

Answer

(i) If 6 pens cost ₹ 69, then the cost of 16 pens is ₹ 184.

(ii) If 1 dozen eggs cost ₹ 54, then a tray of 30 eggs will cost ₹ 135.

(iii) A worker is paid ₹ 1610 as wages for 14 days. His wages for 30 days will be ₹ 3450.

(iv) 25 boxes of 12 ice-cream cups each, cost ₹ 10500. The cost of 15 boxes of 20 ice-cream cups each, will be ₹ 10500.

Explaination

(i) Given:

Cost of 6 pens = ₹ 69

Cost of 1 pen = ₹ $\Big( \dfrac{69}{6} \Big) \quad \text{[Less pens, Less cost]}$

Cost of 16 pens = ₹ $\Big( \dfrac{69}{6} \times 16 \Big) \quad \text{[More pens, More cost]}$

= ₹ $\Big( \dfrac{23}{2} \times 16 \Big)$

= ₹ 23 x 8 $\quad$ [Dividing 16 and 2 by 2]

= ₹ 184

(ii) Given:

1 dozen = 12 eggs

Cost of 12 eggs = ₹ 54

Cost of 1 egg = ₹ $\Big( \dfrac{54}{12} \Big) \quad \text{[Less eggs, Less cost]}$

Cost of 30 eggs = ₹ $\Big( \dfrac{54}{12} \times 30 \Big) \quad \text{[More eggs, Less More]}$

= ₹ $\Big( \dfrac{54}{2} \times 5 \Big) \quad \text{[Dividing 30 and 12 by 6]}$

= ₹ 27 x 5 $\quad$ [Dividing 54 and 2 by 2]

= ₹ 135

(iii) Given:

Wages for 14 days = ₹ 1610

Wages for 1 day = ₹ $\Big( \dfrac{1610}{14} \Big) \quad \text{[Less days, Less wages]}$

Wages for 30 days = ₹ $\Big( \dfrac{1610}{14} \times 30 \Big) \quad \text{[More days, More wages]}$

= ₹ 115 x 30 $\quad$ [Dividing 1610 and 14 by 14]

= ₹ 3450

(iv) Given:

Let us find the total number of cups first

25 boxes x 12 cups = 300 cups

Cost of 300 cups = ₹ 10500

Cost of 1 cup = ₹ $\dfrac{10500}{300} = ₹ 35 \quad \text{[Less cups, Less cost]}$

15 boxes x 20 cups = 300 cups

Cost of 300 cups = ₹ 35 x 300 $\quad$ [More cups, More cost]

= ₹ 10500

Writer true (T) or false (F) :

(i) 12 kg apples for ₹ 2160 is a better buy than 15 kg apples for ₹ 2850.

(ii) If 7 men can finish a work in 84 days, then 12 men can finish the same work in 96 days.

(iii) If 9 notebooks cost ₹ 315, then the cost of 20 notebooks is ₹ 700.

(iv) In an indirect proportion, a decrease in one quantity causes an increase in the other quantity.

Answer

(i) True
Reason — Compare cost per kg

12 kg for ₹2160:

Cost of 1 kg = 2160 ÷ 12 = ₹ 180 per kg

Now, 15 kg for ₹2850:

Cost of 1 kg = 2850 ÷ 15 = ₹ 190 per kg

Since ₹ 180 is cheaper than ₹ 190, the first option is indeed a better buy.

(ii) False
Reason — Days for 7 men = 84 days

Days for 1 man = (84 x 7) days = 588 days $\quad \text{[Less men, More days]}$

Days for 12 men = 588 ÷ 12 = 49 days $\quad \text{[More men, Less days]}$

The statement says 96 days, which is mathematically impossible because increasing the number of men must decrease the time taken.

(iii) True
Reason — Cost of 9 notebook = ₹ 315

Cost of 1 notebook = ₹ $\dfrac{315}{9} = ₹ 35 \quad \text{[Less notebooks, Less cost]}$

Cost of 20 notebooks = ₹ 35 x 20 = ₹ 700 $\quad$[More notebooks, More cost]

The calculation matches the statement perfectly.

(iv) True
Reason — This is the fundamental definition of Indirect (Inverse) Proportion. As one value goes down, the related value must go up to maintain the constant product (x x y = k).

Rajan runs a typing company that processes the manuscript obtained from publishing companies. There are two processes in his work - typing and typesetting. He owns a team each for the two processes. He received a manuscript from a publishing company RBC. He knows that his team of 25 typists can type 225 pages in a day. Also, his team of 7 typesetters can typeset 90 pages in 5 days.

(1) If the manuscript from RBC needs 1350 pages to be typed, how many days will Rajan's team take to type it ?

1. 5
2. 6
3. 7
4. 9

(2) If Rajan employs 5 more typists in his team, in how many days can the work of RBC be completed ?

1. 4
2. 5
3. 6
4. 8

(3) In how many days will Rajan’s team of typesetters typeset the work from RBC ?

1. 50
2. 60
3. 75
4. 85

(4) If typesetting work begins only after the typing work is finished, what is the least number of days Rajan needs to get the work from RBC done by his existing team of 30 typists and 7 typesetters ?

1. 75
2. 80
3. 81
4. 85

Answer

(1) Given:

Number of typists = 25

Pages typed in 1 day = 225

Total manuscript pages to be typed = 1350

Days taken to type 1350 pages = $\dfrac{\text{Total pages}}{\text{Pages typed in 1 day}}$

Substituting the values in above, we get:

Days taken to type 1350 pages = $\dfrac{1350}{225}$

Days taken to type 1350 pages = 6 days

Hence, option 2 is the correct option.

(2) Given:

Original typists = 25

New typists added = 5

Total typists = 25 + 5 = 30

Typing rate of 25 typists = 225 pages per day

Rate of 1 typist = $\dfrac{225}{25} = 9$ pages per day

Rate of 30 typists = 30 x 9 = 270 pages per day

Days taken for 1350 pages = $\dfrac{\text{Total pages}}{\text{Pages typed in 1 day}}$

Substituting the values in above, we get:

Days taken for 1350 pages = $\dfrac{1350}{270}$

Days taken for 1350 pages = 5 days

Hence, option 2 is the correct option.

(3) Given:

Number of typesetters = 7

Pages typeset = 90

Time taken = 5 days

Total manuscript pages = 1350

Pages typeset by 7 typesetters in 1 day = $\dfrac{90}{5}$ = 18 pages

Days taken for 1350 pages = $\dfrac{\text{Total pages}}{\text{Pages typeset in 1 day}}$

Substituting the values in above, we get:

Days taken for 1350 pages = $\dfrac{1350}{18}$

Days taken for 1350 pages = 75 days

Hence, option 3 is the correct option.

(4) Given:

Time taken by 30 typists = 5 days $\quad$[From step 2]

Time taken by 7 typesetters = 75 days $\quad$[From step 3]

Condition: Typesetting starts after typing finishes.

Total days = Typing days + Typesetting days

= 5 + 75

= 80 days

Hence, option 2 is the correct option.

Assertion: When x and y are in indirect proportion, then (x + 1) and (y + 1) are also in indirect proportion.

Reason: Two quantities x and y are said to be in indirect proportion, if xy = k, where k is a constant.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Assertion (A) is false but Reason (R) is true.

Explanation

Let's test Assertion with numbers. If x = 2 and y = 6 (where xy = 12) and then x = 3 and y = 4 (where xy = 12), they are in indirect proportion.

Now add 1: (2+1) = 3 and (6+1) = 7. Here, 3 x 7 = 21.

Next set: (3+1) = 4 and (4+1) = 5. Here, 4 x 5 = 20.

Since $21 \neq 20$, the product is not constant. Therefore, (x+1) and (y+1) are not in indirect proportion. Assertion is False.

Hence, option 4 is the correct option.

Assertion: When the speed is kept fixed, time and distance are in direct proportion.

Reason: Two quantities are said to be in direct proportion if the increase (decrease) in one quantity causes the increase (decrease) in the other quantity.

1. Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).
2. Both Assertion (A) and Reason (R) are true but Reason (R) is not the correct explanation of Assertion (A).
3. Assertion (A) is true but Reason (R) is false.
4. Assertion (A) is false but Reason (R) is true.

Answer

Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of Assertion (A).

Explanation

The formula for distance is Distance = Speed x Time. If speed is constant, doubling the time will exactly double the distance. Thus, they are in direct proportion ($\dfrac{D}{T} = \text{Speed}$). Assertion is True.

Reason is the fundamental definition of direct proportion. Reason is True.

The reason explains why the assertion is true: because as you spend more time traveling at a fixed speed, your distance increases accordingly.

Hence, option 1 is the correct option.