Unitary Method - Exercise 8(B)

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

[Less men, More days]

Days taken by 36 men = $\Big(\dfrac{60 \times 15}{36}\Big) \text{ days}$

[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 = $\dfrac{42}{150} \text{ women}$

[Less cloth, Less women]

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

[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 = $\dfrac{46.8}{1.5} \text{ km}$

[Less cm, Less distance]

Distance for 3.5 cm = $\Big(\dfrac{46.8}{1.5} \times 3.5\Big) \text{ km}$

[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 = $\dfrac{224}{20} \text{ km}$

[Less petrol, Less distance]

Distance on 32.5 L = $\Big(\dfrac{224}{20} \times 32.5\Big) \text{ km}$

[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 = $\dfrac{16}{28} \text{ Buffaloes}$

[Less cows, Less buffaloes]

Buffaloes for 91 cows = $\Big(\dfrac{16}{28} \times 91\Big) \text{ Buffaloes}$

[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 = ₹ $\Big( \dfrac{69}{6} \Big)$

[Less pens, Less cost]

Cost of 16 pens = ₹ $\Big( \dfrac{69}{6} \times 16 \Big)$

[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 = ₹ $\Big( \dfrac{54}{12} \times 30 \Big)$

[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 = ₹ $\Big( \dfrac{1610}{14} \Big)$

[Less days, Less wages]

Wages for 30 days = ₹ $\Big( \dfrac{1610}{14} \times 30 \Big)$

[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 = ₹ $\dfrac{10500}{300} = ₹ 35$

[Less cups, Less cost]

15 boxes x 20 cups = 300 cups

Cost of 300 cups = ₹ 35 x 300

[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 = ₹ $\dfrac{315}{9} = ₹ 35$

[Less notebooks, Less cost]

Cost of 20 notebooks = ₹ 35 x 20 = ₹ 700

[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.

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

Given:

16 buffaloes eat as much as 28 cows.

This is a case of direct proportion - more cows will be equivalent to more buffaloes.

Step 1: Find the number of buffaloes that eat as much as 1 cow

28 cows eat as much as 16 buffaloes.

1 cow eats as much as $\dfrac{16}{28}$ buffaloes. $\quad$[Less cows, Less buffaloes]

Step 2: Find the number of buffaloes that eat as much as 91 cows

91 cows eat as much as $\left(\dfrac{16}{28} \times 91\right)$ buffaloes. $\quad$[More cows, More buffaloes]

= $\dfrac{16 \times 91}{28}$

= $\dfrac{1456}{28}$

= 52

∴ 52 buffaloes eat as much as 91 cows.

Hence, option 2 is the correct option.

Monika received a salary of ₹18,200 for February 2014. How many days will she have to work to get another ₹14,950?

1. 19
2. 18
3. 23
4. 26

Answer

Given:

Salary received by Monika for February 2014 = ₹18,200

Number of days in February 2014 = 28

[∵ 2014 is not a leap year]

This is a case of direct proportion - more days of work, more salary earned.

Step 1: Find the salary for 1 day

Salary for 28 days = ₹18,200

Salary for 1 day = ₹$\dfrac{18200}{28}$

[Less days, Less salary]

= ₹650

Step 2: Find the number of days to earn ₹14,950

Salary for 1 day = ₹650

Number of days to earn ₹14,950 = $\dfrac{14950}{650}$

[More salary, More days]

= 23 days

∴ Monika will have to work for 23 days to earn ₹14,950.

Hence, option 3 is the correct option.

Amit took a loan of ₹500 from a local moneylender, and Kabeer took the same amount from a cooperative society. Amit paid ₹11 every day for 55 days to repay the loan. Kabeer paid ₹600 to the bank after 55 days. Both the loans were closed by 55 days. Who has paid more amount and by how much?

1. Kabeer; ₹5 more than that of Amit
2. Amit; ₹5 more than that of Kabeer
3. Kabeer; ₹10 more than that of Amit
4. None of the above

Answer

Given:

Loan amount taken by Amit = ₹500

Daily repayment by Amit = ₹11

Number of days for which Amit paid = 55

Amount paid by Kabeer after 55 days = ₹600

Step 1: Find the total amount paid by Amit

Amit paid ₹11 every day for 55 days.

Total amount paid by Amit = 11 × 55

= ₹605

Step 2: Compare the amounts paid by Amit and Kabeer

Amount paid by Amit = ₹605

Amount paid by Kabeer = ₹600

Difference = 605 − 600 = ₹5

Since ₹605 > ₹600, Amit paid ₹5 more than Kabeer.

∴ Amit paid ₹5 more than Kabeer.

Hence, option 2 is the correct option.

In a 50-over game of cricket, Team A scored only 3.6 runs per over in the first 15 overs. What should be the runs per over in the remaining 35 overs to reach the target of 299 runs?

1. 6.75
2. 7.5
3. 7
4. 6.5

Answer

Given:

Total overs in the game = 50

Runs per over in the first 15 overs = 3.6

Target = 299 runs

Step 1: Find the runs scored in the first 15 overs

Runs scored in 15 overs = 3.6 × 15

= 54 runs

Step 2: Find the runs needed in the remaining overs

Remaining runs = Target − Runs scored in first 15 overs

= 299 − 54

= 245 runs

Remaining overs = 50 − 15 = 35 overs

Step 3: Find the required runs per over

Required runs per over = $\dfrac{\text{Remaining runs}}{\text{Remaining overs}}$

= $\dfrac{245}{35}$

= 7 runs per over

∴ Team A should score 7 runs per over in the remaining 35 overs to reach the target.

Hence, option 3 is the correct option.

The cost of 2 dozen bananas is ₹240, and the cost of 18 lemons is ₹90. The ratio of the cost of 1 lemon to that of 1 banana is:

1. 1 : 2
2. 2 : 3
3. 3 : 2
4. 2 : 1

Answer

Given:

Cost of 2 dozen bananas = ₹240

Cost of 18 lemons = ₹90

1 dozen = 12, so 2 dozen = 24 bananas.

Step 1: Find the cost of 1 banana

Cost of 24 bananas = ₹240

Cost of 1 banana = ₹$\dfrac{240}{24}$

[Less bananas, Less cost]

= ₹10

Step 2: Find the cost of 1 lemon

Cost of 18 lemons = ₹90

Cost of 1 lemon = ₹$\dfrac{90}{18}$

[Less lemons, Less cost]

= ₹5

Step 3: Find the ratio of the cost of 1 lemon to that of 1 banana

Ratio = Cost of 1 lemon : Cost of 1 banana

= 5 : 10

$= \dfrac{5}{10} \quad \text{[Writing the ratio as a fraction]} \\[1em] = \dfrac{1}{2} \\[1em] = 1 : 2$

∴ The ratio of the cost of 1 lemon to that of 1 banana is 1 : 2.

Hence, option 1 is the correct option.

At a particular time of a day, a 7 m high flagstaff casts a shadow which is 8.2 m long. What is the height of the building which casts a shadow of 20.5 m in length at the same moment?

1. 175 m
2. 17.5 m
3. 1.75 m
4. None of these

Answer

Given:

Height of flagstaff = 7 m

Length of shadow of flagstaff = 8.2 m

Length of shadow of building = 20.5 m

This is a case of direct proportion - more shadow length, more height.

Step 1: Find the height corresponding to 1 m shadow

For 8.2 m shadow, height = 7 m

For 1 m shadow, height = $\dfrac{7}{8.2}$ m

[Less shadow, Less height]

Step 2: Find the height of the building corresponding to 20.5 m shadow

For 20.5 m shadow, height = $\left(\dfrac{7}{8.2} \times 20.5\right)$ m

[More shadow, More height]

= $\dfrac{7 \times 20.5}{8.2}$

= $\dfrac{143.5}{8.2}$

= 17.5 m

∴ The height of the building is 17.5 m.

Hence, option 2 is the correct option.

Ruchi types 40 words per minute and takes 24 minutes to type a certain document. Her friend Geeta has a typing speed of 48 words per minute. In how much time will she be able to type the same document?

1. 10 minutes
2. 15 minutes
3. 20 minutes
4. 25 minutes

Answer

Given:

Ruchi's typing speed = 40 words per minute

Time taken by Ruchi = 24 minutes

Geeta's typing speed = 48 words per minute

This is a case of indirect proportion - more typing speed, less time taken to type the same document.

Step 1: Find the total number of words in the document

Total words = Typing speed × Time taken

= 40 × 24

= 960 words

Step 2: Find the time taken by Geeta to type the document

Geeta types 48 words in 1 minute.

Geeta types 1 word in $\dfrac{1}{48}$ minute. $\quad$[Less words, Less time]

Geeta types 960 words in $\left(\dfrac{1}{48} \times 960\right)$ minutes. $\quad$[More words, More time]

= $\dfrac{960}{48}$

= 20 minutes

∴ Geeta will take 20 minutes to type the same document.

Hence, option 3 is the correct option.