Q1. Cost of 24 identical articles is Rs.108. Find cost of 40 identical articles.
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$\Rightarrow$ Direct relation between no. of articles and its cost. More articles will cost more and vice-versa.
$\Rightarrow$ 24 articles cost = $ Rs.108$
$\Rightarrow$ 1 article costs = $\dfrac{108}{24}$
$\therefore$ 40 articles cost = $\dfrac{108}{24} \times 40$
$\Rightarrow \dfrac{108}{6} \times 10 = 18 \times 10 = Rs.180$ Answer
$\Rightarrow$ 24 articles cost = $ Rs.108$
$\Rightarrow$ 1 article costs = $\dfrac{108}{24}$
$\therefore$ 40 articles cost = $\dfrac{108}{24} \times 40$
$\Rightarrow \dfrac{108}{6} \times 10 = 18 \times 10 = Rs.180$ Answer
Q2. If 15 men can complete a peice of work in 30 days, in how many days will 18 men complete it?
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$\Rightarrow$ Inverse relation between no. of men and days. More men will take less days and less men will take more days.
$\Rightarrow$ 15 men finish a work in = $30$ days
$\Rightarrow$ 1 man will finish the work in = $30 \times 15$
$\Rightarrow$ 18 men will finish the work in = $\dfrac{30 \times 15}{18}$
$\Rightarrow \dfrac{5 \times 15}{3} = 5 \times 5 = 25$ days Answer
$\Rightarrow$ 15 men finish a work in = $30$ days
$\Rightarrow$ 1 man will finish the work in = $30 \times 15$
$\Rightarrow$ 18 men will finish the work in = $\dfrac{30 \times 15}{18}$
$\Rightarrow \dfrac{5 \times 15}{3} = 5 \times 5 = 25$ days Answer
Q3. In order to complete a work in 28 days, 60 men are required. How many men will be required if the same work is to be completed in 40 days?
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$\Rightarrow$ No. of men finish a work in 28 days = $60$
$\Rightarrow$ No. of men finish a work in 1 day = $60 \times 28$
$\Rightarrow$ No. of men finish a work in 40 days = $\dfrac{60 \times 28}{40}$
$\Rightarrow \dfrac{6 \times 28}{4} = 6 \times 7 = 42$ men Answer
$\Rightarrow$ No. of men finish a work in 1 day = $60 \times 28$
$\Rightarrow$ No. of men finish a work in 40 days = $\dfrac{60 \times 28}{40}$
$\Rightarrow \dfrac{6 \times 28}{4} = 6 \times 7 = 42$ men Answer
Q4. A fort has provisions for 450 soldiers for 40 days. After 10 days, 90 more soldiers came to the fort. Find in how many days will the remaining provisions last at the same rate?
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$\Rightarrow$ Inverse relation between no. of soldiers and days. More soldiers will have food for less days and less men will have food for more days.
$\Rightarrow$ New no. of soldiers = $450 + 90 = 540$
$\Rightarrow$ No. of days provisions will last for 450 soldiers = $30$
$\Rightarrow$ No. of days provisions will last for 1 soldier = $30 \times 450$
$\Rightarrow$ No. of days provisions will last for 540 soldier = $\dfrac{30 \times 450}{540}$
$\Rightarrow \dfrac{30 \times 5}{6} = 5 \times 5 = 25$ days Answer
$\Rightarrow$ New no. of soldiers = $450 + 90 = 540$
$\Rightarrow$ No. of days provisions will last for 450 soldiers = $30$
$\Rightarrow$ No. of days provisions will last for 1 soldier = $30 \times 450$
$\Rightarrow$ No. of days provisions will last for 540 soldier = $\dfrac{30 \times 450}{540}$
$\Rightarrow \dfrac{30 \times 5}{6} = 5 \times 5 = 25$ days Answer
Q5. A garrison has sufficient provisions for 480 men for 12 days. If the number of men is reduced by 160; find long will the provisions last?
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$\Rightarrow$ Inverse relation between no. of soldiers and days. More soldiers will have food for less days and less men will have food for more days.
$\Rightarrow$ New no. of soldiers = $480 + 160 = 320$
$\Rightarrow$ No. of days provisions will last for 480 soldiers = $12$
$\Rightarrow$ No. of days provisions will last for 1 soldier = $12 \times 480$
$\Rightarrow$ No. of days provisions will last for 320 soldier = $\dfrac{12 \times 480}{320}$
$\Rightarrow \dfrac{12 \times 48}{32} = \dfrac{12 \times 6}{4} = 3 \times 6 = 18$ days Answer
$\Rightarrow$ New no. of soldiers = $480 + 160 = 320$
$\Rightarrow$ No. of days provisions will last for 480 soldiers = $12$
$\Rightarrow$ No. of days provisions will last for 1 soldier = $12 \times 480$
$\Rightarrow$ No. of days provisions will last for 320 soldier = $\dfrac{12 \times 480}{320}$
$\Rightarrow \dfrac{12 \times 48}{32} = \dfrac{12 \times 6}{4} = 3 \times 6 = 18$ days Answer
Q6. $\dfrac{3}{5}$ quintal of wheat costs Rs.210. Find the cost of:
(i) 1 quintal of wheat
(ii) 0.4 quintal of wheat
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(i) Direct relation between quintal and cost. More quintals will cost more and vice-versa.
$\Rightarrow $ 1 quintal = $100 kg $
$\Rightarrow \dfrac{3}{5}$ quintal = $\dfrac{3}{5} \times 100 = 3 \times 20 = 60 kg$
$\Rightarrow $ cost of 60 kg wheat = $ Rs.210$
$\Rightarrow $ cost of 1 kg wheat = $\dfrac{210}{60}$
$\Rightarrow $ cost of 100 kg (1 quintal) wheat = $\dfrac{210}{60} \times 100$
$\Rightarrow \dfrac{7}{2} \times 100 = 7 \times 50$ = Rs.350 Answer
(ii) 0.4 quintal = $\dfrac{4}{10} \times 100$
$\Rightarrow 4 \times 10 = 40$ kg
$\Rightarrow $ cost of 1 kg wheat = $\dfrac{210}{60} = \dfrac{7}{2} = 3.5$
$\Rightarrow $ cost of 40 kg wheat = $3.5 \times 40$
$\Rightarrow \dfrac{35}{10} \times 40 = 35 \times 4 = Rs.140$ Answer
$\Rightarrow $ 1 quintal = $100 kg $
$\Rightarrow \dfrac{3}{5}$ quintal = $\dfrac{3}{5} \times 100 = 3 \times 20 = 60 kg$
$\Rightarrow $ cost of 60 kg wheat = $ Rs.210$
$\Rightarrow $ cost of 1 kg wheat = $\dfrac{210}{60}$
$\Rightarrow $ cost of 100 kg (1 quintal) wheat = $\dfrac{210}{60} \times 100$
$\Rightarrow \dfrac{7}{2} \times 100 = 7 \times 50$ = Rs.350 Answer
(ii) 0.4 quintal = $\dfrac{4}{10} \times 100$
$\Rightarrow 4 \times 10 = 40$ kg
$\Rightarrow $ cost of 1 kg wheat = $\dfrac{210}{60} = \dfrac{7}{2} = 3.5$
$\Rightarrow $ cost of 40 kg wheat = $3.5 \times 40$
$\Rightarrow \dfrac{35}{10} \times 40 = 35 \times 4 = Rs.140$ Answer
Remember:
(a) 1 quinal = 100 kg
(b) 1 tonne = 1000kg
(c) 1 kg = 1000g
(d) 1 g = 0.001 kg
(e) 1 g = 1000 mg
(f) 1 mg = 0.001 g
View Metric System
Q7. If $\dfrac{2}{9}$ of a property Rs.2,52,000; find the cost of $\dfrac{4}{7}$ of it.
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$\Rightarrow$ Direct relation between cost of property and its cost. More property costs more and vice-versa.
$\Rightarrow$ cost of $\dfrac{2}{9}$ of a property = $ Rs.2,52,000$
$\Rightarrow$ cost of a property (whole) = $\dfrac{2,52,000}{\dfrac{2}{9}}$
$\Rightarrow \dfrac{2,52,000}{2} \times 9 = 1,26,000 \times 9 = Rs.11,34,000$
$\therefore$ cost of $\dfrac{4}{7}$ of a property = $11,34,000 \times \dfrac{4}{7}$
$\Rightarrow 1,62,000 \times 4 = Rs.6,48,000$ Answer
$\Rightarrow$ cost of $\dfrac{2}{9}$ of a property = $ Rs.2,52,000$
$\Rightarrow$ cost of a property (whole) = $\dfrac{2,52,000}{\dfrac{2}{9}}$
$\Rightarrow \dfrac{2,52,000}{2} \times 9 = 1,26,000 \times 9 = Rs.11,34,000$
$\therefore$ cost of $\dfrac{4}{7}$ of a property = $11,34,000 \times \dfrac{4}{7}$
$\Rightarrow 1,62,000 \times 4 = Rs.6,48,000$ Answer
Q8. 4 men or 6 women earn Rs.360 in one day. Find how much will:
(i) a man earn in one day?
(ii) a woman earn in one day?
(iii) 6 men and 4 women earn in one day?
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$\Rightarrow$ Direct relation between no. of people and their earning. More people will earn more and vice-versa.
(i) 4 men earn = $Rs.360$
$\Rightarrow$ 1 man earns = $\dfrac{360}{4} = Rs.90$ Answer
(ii)6 women earn = $Rs.360$
$\Rightarrow$ 1 man earns = $\dfrac{360}{6} = Rs.60$ Answer
(iii) 6 men and 4 women will earn = $(6 \times 90) + (4 \times 60)$
$\Rightarrow = 540 + 240 = Rs.780$ Answer
(i) 4 men earn = $Rs.360$
$\Rightarrow$ 1 man earns = $\dfrac{360}{4} = Rs.90$ Answer
(ii)6 women earn = $Rs.360$
$\Rightarrow$ 1 man earns = $\dfrac{360}{6} = Rs.60$ Answer
(iii) 6 men and 4 women will earn = $(6 \times 90) + (4 \times 60)$
$\Rightarrow = 540 + 240 = Rs.780$ Answer
Q9. 16 boys went to canteen to have tea and snacks together. The bill amounted to Rs.114.40. What will be the contribution of a boy who pays for himself and 5 others?
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$\Rightarrow$ Direct relation between no. of boys and their bill amount. More no. of boys will generate more bill amount and vice-versa.
$\Rightarrow$ bill of 16 boys = $ Rs.114.40$
$\Rightarrow$ bill of 1 boy = $\dfrac{114.40}{16} = Rs.7.15$
$\Rightarrow$ bill of 6 boys = $7.15 \times 6 = Rs.42.90$ Answer
$\Rightarrow$ bill of 16 boys = $ Rs.114.40$
$\Rightarrow$ bill of 1 boy = $\dfrac{114.40}{16} = Rs.7.15$
$\Rightarrow$ bill of 6 boys = $7.15 \times 6 = Rs.42.90$ Answer
Q10. 50 labourers can dig a pond in 16 days. How many labourers will be required to dig another pond, double in size, in 30 days?
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$\Rightarrow$ Inverse relation between no. of labourers and days. More labourers will take less days and vice-versa.
$\Rightarrow$ No. of labourers take to dig a pond in 16 days = $50$
$\Rightarrow$ No. of labourers take to dig a pond in 1 day = $16 \times 50 = 800$
$\therefore$ No. of labourers take to dig a pond in 20 days = $\dfrac{800}{20} = 40$
$\therefore$ Labourers required to dig a pond double in size = $40 \times 2 = 80$ days Answer
$\Rightarrow$ No. of labourers take to dig a pond in 16 days = $50$
$\Rightarrow$ No. of labourers take to dig a pond in 1 day = $16 \times 50 = 800$
$\therefore$ No. of labourers take to dig a pond in 20 days = $\dfrac{800}{20} = 40$
$\therefore$ Labourers required to dig a pond double in size = $40 \times 2 = 80$ days Answer
Q11. If 12 men or 16 women can complete a piece of work in 7 days, in how many days can 4 men and 8 women complete the same work?
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$\Rightarrow$ Direct relation between no. of men and women. More men means more women and vice-versa.
$\Rightarrow$ 12 men = $18$ women
$\Rightarrow$ 1 man = $\dfrac{18}{12}$
$\therefore$ 4 men = $\dfrac{18}{12} \times 4 = \dfrac{18}{3} = 6$ women
$\Rightarrow$ 4 men and 8 women = $6 + 8 = 14$ women
$\Rightarrow$ No. of days 18 women take to do a piece of work = $7$ days
$\Rightarrow$ No. of days 1 woman will take to do a piece of work = $7 \times 18 = 126$
$\Rightarrow$ No. of days 14 women will take to do a piece of work = $\dfrac{126}{14} = 9$ days Answer
$\Rightarrow$ 12 men = $18$ women
$\Rightarrow$ 1 man = $\dfrac{18}{12}$
$\therefore$ 4 men = $\dfrac{18}{12} \times 4 = \dfrac{18}{3} = 6$ women
$\Rightarrow$ 4 men and 8 women = $6 + 8 = 14$ women
$\Rightarrow$ No. of days 18 women take to do a piece of work = $7$ days
$\Rightarrow$ No. of days 1 woman will take to do a piece of work = $7 \times 18 = 126$
$\Rightarrow$ No. of days 14 women will take to do a piece of work = $\dfrac{126}{14} = 9$ days Answer
Q12. If 3 men or 6 boys can finish a work in 20 days, how long will 4 men and 12 boys take to finish the same work?
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$\Rightarrow$ Direct relation between no. of men and women. More men means more women and vice-versa.
$\Rightarrow$ 3 men = $6$ boys
$\Rightarrow$ 1 man = $\dfrac{6}{3} = 2$ boys
$\therefore$ 4 men = $2 \times 4 = 8$ boys
$\Rightarrow$ 4 men and 12 boys = $8 + 12 = 20$ boys
$\Rightarrow$ No. of days 6 boys take to finish a work = $20$ days
$\Rightarrow$ No. of days 1 boy will take to finish a work = $20 \times 6 = 120$ days
$\Rightarrow$ No. of days 20 boys will take to finish a work = $\dfrac{120}{20} = 6$ days Answer
$\Rightarrow$ 3 men = $6$ boys
$\Rightarrow$ 1 man = $\dfrac{6}{3} = 2$ boys
$\therefore$ 4 men = $2 \times 4 = 8$ boys
$\Rightarrow$ 4 men and 12 boys = $8 + 12 = 20$ boys
$\Rightarrow$ No. of days 6 boys take to finish a work = $20$ days
$\Rightarrow$ No. of days 1 boy will take to finish a work = $20 \times 6 = 120$ days
$\Rightarrow$ No. of days 20 boys will take to finish a work = $\dfrac{120}{20} = 6$ days Answer
Q13. A particular work can be completed by 6 men and 6 women in 24 days; whereas the same work can be completed by 8 men and 12 women in 15 days. Find:
(i) according to the amount of work done, one man is equivalent to how many women.
(ii) the time taken by 4 men and 6 women to complete the same work.
Show Answer
$\Rightarrow$ Inverse relation between no. of people and days. More people will take less days and vice-versa.
$\Rightarrow$ No. of people required to complete a work in 24 days = 6 men + 6 women
$\Rightarrow$ No. of people required to complete a work in 1 day = 24(6 men + 6 women)
$\Rightarrow$ 144 men + 144 women
Similarly:
$\Rightarrow$ No. of people required to complete a work in 15 days = 8 men + 12 women
$\Rightarrow$ No. of people required to complete a work in 1 day = 15(8 men + 12 women)
$\Rightarrow$ 120 men + 180 women
(i) Now, 144 men + 144 women = 120 men + 180 women
$\Rightarrow$ 144 men - 180 women = 120 men - 144 women
$\Rightarrow$ 24 men = 36 women
$\Rightarrow$ 1 man = $\dfrac{36}{24} = \dfrac{3}{2}$ or $1\dfrac{1}{2}$ women Answer
Now:
$\because$ 6 men and 6 women = ($6 \times \dfrac{3}{2}$) + 6 women
$\Rightarrow 9 + 6 = 15$ women
Similarly:
$\because$ 4 men and 6 women = ($4 \times \dfrac{3}{2}) + 6$ women
$\Rightarrow 6 + 6 = 12$ women
(ii) No. of days required to complete the work by 15 women = 24 days
$\Rightarrow$ No. of days required to complete the work by 1 woman = 24 $\times$ 15
$\Rightarrow$ No. of days required to complete the work by 12 women = $\dfrac{24 \times 15}{12} = 2 \times 15 = 30$ days Answer
$\Rightarrow$ No. of people required to complete a work in 24 days = 6 men + 6 women
$\Rightarrow$ No. of people required to complete a work in 1 day = 24(6 men + 6 women)
$\Rightarrow$ 144 men + 144 women
Similarly:
$\Rightarrow$ No. of people required to complete a work in 15 days = 8 men + 12 women
$\Rightarrow$ No. of people required to complete a work in 1 day = 15(8 men + 12 women)
$\Rightarrow$ 120 men + 180 women
(i) Now, 144 men + 144 women = 120 men + 180 women
$\Rightarrow$ 144 men - 180 women = 120 men - 144 women
$\Rightarrow$ 24 men = 36 women
$\Rightarrow$ 1 man = $\dfrac{36}{24} = \dfrac{3}{2}$ or $1\dfrac{1}{2}$ women Answer
Now:
$\because$ 6 men and 6 women = ($6 \times \dfrac{3}{2}$) + 6 women
$\Rightarrow 9 + 6 = 15$ women
Similarly:
$\because$ 4 men and 6 women = ($4 \times \dfrac{3}{2}) + 6$ women
$\Rightarrow 6 + 6 = 12$ women
(ii) No. of days required to complete the work by 15 women = 24 days
$\Rightarrow$ No. of days required to complete the work by 1 woman = 24 $\times$ 15
$\Rightarrow$ No. of days required to complete the work by 12 women = $\dfrac{24 \times 15}{12} = 2 \times 15 = 30$ days Answer
Q14. If 12 men and 16 boys can do a piece of work in 5 days and, 13 men and 24 boys can it in 4 days, how long will 7 men and 10 boys take to do it?
Show Answer
$\Rightarrow$ Inverse relation between no. of people and days. More people will take less days and vice-versa.
$\Rightarrow$ No. of people required to complete a work in 5 days = 12 men + 16 boys
$\Rightarrow$ No. of people required to complete a work in 1 day = 5(12 men + 16 boys)
$\Rightarrow$ 60 men + 80 boys
Similarly:
$\Rightarrow$ No. of people required to complete a work in 4 days = 13 men + 24 boys
$\Rightarrow$ No. of people required to complete a work in 1 day = 4(13 men + 24 boys)
$\Rightarrow$ 52 men + 96 boys
So we have the following equation:
$\Rightarrow$ 60 men + 80 boys = 52 men + 96 boys
$\Rightarrow$ 60 men - 52 men = 96 boys - 80 boys
$\Rightarrow$ 8 men = 16 boys
$\Rightarrow$ 1 man = $\dfrac{16}{8} = 2$ boys
Now:
$\because$ 12 men + 16 boys = $(12 \times 2) + 16 = 40$ boys
and 7 men + 10 boys = $(7 \times 2) + 10 = 24$ boys
$\therefore$ No. of days required to complete the work by 40 boys = 5 days
$\Rightarrow$ No. of days required to complete the work by 1 boy = $5 \times 40 = 200$ days
$\Rightarrow$ No. of days required to complete the work by 24 boys = $\dfrac{200}{24} = \dfrac{25}{3} = 8\dfrac{1}{3}$ days Answer
$\Rightarrow$ No. of people required to complete a work in 5 days = 12 men + 16 boys
$\Rightarrow$ No. of people required to complete a work in 1 day = 5(12 men + 16 boys)
$\Rightarrow$ 60 men + 80 boys
Similarly:
$\Rightarrow$ No. of people required to complete a work in 4 days = 13 men + 24 boys
$\Rightarrow$ No. of people required to complete a work in 1 day = 4(13 men + 24 boys)
$\Rightarrow$ 52 men + 96 boys
So we have the following equation:
$\Rightarrow$ 60 men + 80 boys = 52 men + 96 boys
$\Rightarrow$ 60 men - 52 men = 96 boys - 80 boys
$\Rightarrow$ 8 men = 16 boys
$\Rightarrow$ 1 man = $\dfrac{16}{8} = 2$ boys
Now:
$\because$ 12 men + 16 boys = $(12 \times 2) + 16 = 40$ boys
and 7 men + 10 boys = $(7 \times 2) + 10 = 24$ boys
$\therefore$ No. of days required to complete the work by 40 boys = 5 days
$\Rightarrow$ No. of days required to complete the work by 1 boy = $5 \times 40 = 200$ days
$\Rightarrow$ No. of days required to complete the work by 24 boys = $\dfrac{200}{24} = \dfrac{25}{3} = 8\dfrac{1}{3}$ days Answer
Arrow Method
Q15. 8 oranges can be bought for Rs.10.40. How many more can be bought for Rs.16.90?
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$\Rightarrow$ $$ \left[
\begin{array}{ c c c }
Amount & & Oranges\\
10.40 & \downarrow & 8\\
& \downarrow & \\
16.90 & \downarrow & x \\
& Direct\,Relation & \end{array} \right] $$
$\Rightarrow \dfrac{x}{8}$ = $\dfrac{16.90}{10.40}$
$\Rightarrow$ $x$ = $\dfrac{1690 \times 8 \times 100}{1040 \times 100}$
$\Rightarrow$ $x$ = $\dfrac{1690 \times 8}{1040}$
$\Rightarrow$ $x$ = $\dfrac{1690}{130}$
$\Rightarrow$ $x$ = 13
$\therefore$ Required no. of oranges = $13 - 8 = 5$ Answer
$\Rightarrow \dfrac{x}{8}$ = $\dfrac{16.90}{10.40}$
$\Rightarrow$ $x$ = $\dfrac{1690 \times 8 \times 100}{1040 \times 100}$
$\Rightarrow$ $x$ = $\dfrac{1690 \times 8}{1040}$
$\Rightarrow$ $x$ = $\dfrac{1690}{130}$
$\Rightarrow$ $x$ = 13
$\therefore$ Required no. of oranges = $13 - 8 = 5$ Answer