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MATHS [4 marks sum]

Unitary Method (Including Time and Work) · ICSE STD 7 (ICSE - English Medium). 15 questions.

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[4 marks sum]

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15 questions · timed · auto-graded

Question 14 Marks
A is able to complete $\frac{1}{3}$ of a certain work in 10 hrs and B is able to complete $\frac{2}{5}$ of the same work in 12 hrs. Find:
(i) how much work can $\mathrm{A}$ do in 1 hour?
(ii) how much work can B do in 1 hour?
(iii) in how much time will the work be completed, if both work together.
Answer
A can do $\frac{1}{3}$ of work in $=10$ hours
$\therefore$ A can do full work in $=\frac{10 \times 3}{1}=30$ hours
B can do $\frac{2}{5}$ of the work in $=12$ hours
$\therefore B$ can do the whole work in $=\frac{12 \times 5}{2}=30$ hours
(i) Now A's 1 hour's work $=\frac{1}{30}$
(ii) B's 1 hours work $=\frac{1}{30}$
(iii) Both's 1 hour's work
$ =\frac{1}{30}+\frac{1}{30}=\frac{2}{30}=\frac{1}{15} $
$\therefore$ Both can finish the work in 15 hours
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Question 24 Marks
Joseph and Peter can complete a work in 20 hours and 25 hours respectively. Find :
(i) work done by both together in 4 hrs.
(ii) work left after both worked together for 4 hrs.
(iii) time taken by Peter to complete the remaining work.
Answer
Joseph can do a work in $=20$ hours
Peter can do the same work in $=25$ hours
Now Joseph's 1 hour's work $=\frac{1}{20}$
and Peter's 1 hour's work $=\frac{1}{25}$
Both's 1 hour's work $=\frac{1}{20}+\frac{1}{25}=\frac{5+4}{100}=\frac{9}{100}$
(i) Both's 4 hours work $=\frac{9}{100} \times 4=\frac{9}{25}$
(ii) Work left over $=1-\frac{9}{25}$
$ =\frac{25-9}{25}=\frac{16}{25} $
(iii) Peter can do $\frac{16}{25}$ work in $=25 \times \frac{16}{25}=16$ hours
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Question 34 Marks
Mohit can complete a work in 50 days, whereas Anuj can complete the same work in 40 days. Find:
(i) work done by Mohit in 20 days.
(ii) work left after Mohit has worked on it for 20 days.
(iii) time taken by Anuj to complete the remaining work.
Answer
Mohit can complete a work in 50 days
and Anuj can complete the same work in 40 days
$\therefore$ Mohit's one day's work $=\frac{1}{50}$
and Anuj's one day's work $=\frac{1}{40}$
(i) Mohit's 20 day's work $=\frac{1}{50} \times 20=\frac{2}{5}$
(ii) Work left $=1-\frac{2}{5}=\frac{5-2}{5}=\frac{3}{5}$
Anuj can do $\frac{3}{5}$ work in $=40 \times \frac{3}{5}$ days $=24$ days
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Question 44 Marks
Two taps can fill a cistern in 10 hours and 8 hours respectively. A third tap can empty it in 15 hours. How long will it take to fill the empty cistern, if all of them are opened together?
Answer
First tap can fill a cistern in 10 hours
Second tap can fill the cistern in 8 hours
Third tap can empty the cistern in 15 hours
$\therefore$ First tap's 1 hour's work $=\frac{1}{10}$
Second tap's 1 hour's work $=\frac{1}{8}$
and third tap's 1 hour's work $=\frac{1}{15}$
If all of them are opened together, then
heir one hour's work $=\frac{1}{10}+\frac{1}{8}-\frac{1}{15}$
$=\frac{12+15-8}{120}=\frac{27-8}{120}=\frac{19}{120}$
$\therefore$ They can fill the cistren in $=\frac{120}{19}$ hours
$=6 \frac{6}{19}$ hours
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Question 54 Marks
A, B and C can do a piece of work in 12, 15 and 20 days respectively. How long will they take to do it working together?
Answer
A can do a piece of work in 12 days
B can do the same work in 15 days
C can do the same work in 20 days
$\therefore$ A's 1 day's work $=\frac{1}{12}$
B's 1 day's work $=\frac{1}{15}$
C's 1 day's work $=\frac{1}{20}$
$\therefore(A+B+C)$ 's together 1 day's work
$=\frac{1}{12}+\frac{1}{15}+\frac{1}{20}$
$=\frac{5+4+3}{60}=\frac{12}{60}=\frac{1}{5}$
$\therefore$ they can do the work in 5 days.
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Question 64 Marks
A and B take 6 hours and 9 hours respectively to complete a work. A works for 1 hour and then B works for two hours.
(i) How much work is done in these 3 hours?
(ii) How much work is still left?
Answer
A takes 6 hours to finish work
and B take 9 hours to finish the same work
$\therefore$ A's 1 hour's work $=\frac{1}{6}$
and B's 1 hour's work $=\frac{1}{9}$
and B's 2 hours work $=\frac{1}{9} \times 2=\frac{2}{9}$
(i) Now A's 1 hours work + B's 2 hours work
$=\frac{1}{6}+\frac{2}{9}=\frac{3+4}{18}=\frac{7}{18}$
(ii) Work left $=1-\frac{7}{18}=\frac{18-7}{18}=\frac{11}{18}$
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Question 74 Marks
A can do a piece of work in 4 days and B can do the same work in 5 days. Find, how much work can be done by them working together in:
  1. one day
  2. 2 days
What part of work will be left, after they have worked together for 2 days?
Answer
A can do a piece of work in 4 days
and $B$ can do the same work in 5 days
$\therefore$ A's one day's work $=\frac{1}{4}$
and $B ;$ s one day's work $=\frac{1}{5}$
(i) A and B's both one day's work $=\frac{1}{4}+\frac{1}{5}$
$ =\frac{5+4}{20}=\frac{9}{20} $
(ii) A and B's 2 day's work $=\frac{9}{20} \times 2=\frac{9}{10}$
$\therefore$ work left after 2 day's $=1-\frac{9}{10}$
$ =\frac{10-9}{10}=\frac{1}{10} $
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Question 84 Marks
A and B working together can do a piece of work in 10 days B alone can do the same work in 15 days. How long will A alone take to do the same work?
Answer
$A$ and $B$ working together can do a piece of work in 10 days
and $\mathrm{B}$ alone can do the same work in 15 days
A and $B$ 's one day's work $=\frac{1}{10}$
and B's one day's work $=\frac{1}{15}$
$\therefore$ A's one day's work $=\frac{1}{10}-\frac{1}{15}$
$=\frac{3-2}{30}=\frac{1}{30}$
Hence $A$ can do the same work in $=30$ days
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Question 94 Marks
A, B and C together finish a work in 4 days. If A alone can finish the same work in 8 days and B in 12 days, find how long will C take to finish the work.
Answer
A. $B$ and $C$ finish work together in $=4$ days.
A. B and C finish work together in 1 day $=\frac{1}{4}$
$A^{\prime}$ one day work $=\frac{1}{8}$
B's one day work $=\frac{1}{12}$
$\therefore$ C's one day work $=\frac{1}{4}-\left(\frac{1}{8}+\frac{1}{12}\right)$
$=\frac{6-(3+2)}{24}=\frac{1}{24}$
$\therefore C$ can finish the work in $=24$ days
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Question 104 Marks
Shaheed can prepare one wooden chair in 3 days and Shaif can prepare the same chair in 4 days. If they work together, in how many days will they prepare:
(i) one chair?
(ii)14 chairs of the same kind?
Answer
Shaheed's 1 day's work $=\frac{1}{3}$
and Shaif's 1 day's work $=\frac{1}{4}$
Both one day's work $=\frac{1}{3}+\frac{1}{4}=\frac{4+3}{12}=\frac{7}{12}$
$\therefore$ Both can prepare the chair in $=\frac{12}{7}$ days
$=1 \frac{5}{7}$ days
One chair is prepared in $=1 \frac{5}{7}$ days
$\therefore 14$ chairs will be prepared in $=\frac{12}{7} \times 14=24$ days.
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Question 114 Marks
A can do a piece of work in 6 days and B can do it in 8 days. How long will they take to complete it together?
Answer
A can do a work in $=6$ days
$\therefore$ A's one day's work $=\frac{1}{6}$
B can do the same work in $=8$ dyas
$\therefore B$ is one day's work $=\frac{1}{8}$
$\therefore A$ ad B's both one day's work
$ =\frac{1}{6}+\frac{1}{8}=\frac{4+3}{24}=\frac{7}{24} $
$\therefore$ Both $\mathrm{A}$ and $\mathrm{B}$ can do the same work in $\frac{24}{7}=3 \frac{3}{7}$ days
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Question 124 Marks
If 32 apples weigh 2 kg 800 g. How many apples will there be in a box, containing 35 kg of apples?
Answer
Apples in a box $=35 \mathrm{~kg}$
Now, If weight is $2 \mathrm{~kg} 800 \mathrm{~g}=2.800 \mathrm{~kg}$,
then number of apples $=32$
if weight is $1 \mathrm{~kg}$, then number of apples $=\frac{32}{2.800}$
and if weight is $35 \mathrm{~kg}$, the number of apples
$ \begin{aligned} & =\frac{32 \times 35}{2.800} \\ & =\frac{32 \times 35 \times 1000}{2800}=400 \end{aligned} $
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Question 134 Marks
In a school’s hostel mess, 20 children consume a certain quantity of ration in 6 days. However, 5 children did not return to the hostel after holidays. How long will the same amount of ration last now?
Answer
Total number of children $=20$
20 children consume a certain quantity of ration in $=6$ days
1 children consume a certain quantity of ration in $=6 \times 20$ days
As 5 children did not return to the hostel after holidays.
Then number of children in hostel $=20-5=15$
Hence, 15 children consume certain quantity $6 \times 20$
of ration in $=\frac{6 \times 20}{15}$ days $=8$ days
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Question 144 Marks
A machine is used for making rubber balls and makes 500 balls in 30 minutes. How many balls will it make in 3 1/2 hours?
Answer
30 minutes $=\frac{30}{60}=\frac{1}{2}$ hours
In $\frac{1}{2}$ hours ball makes by machine $=500$
In 1 hours balls makes by machine $=500 \times \frac{2}{1}=1000$ balls
$\therefore \ln 3 \frac{1}{2}\left(\frac{7}{2}\right)$ hours, Balls make by machine
$=500 \times 2 \times \frac{7}{2}=3500$ balls
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Question 154 Marks
125 pupils have food sufficient for 18 days. If 25 more pupils join them, how long will the food last now? What assumption have you made to come to your answer?
Answer
Pupils in the beginning $=125$
More pupils joined $=25$
Total pupils $=125+25=150$
Food is sufficient for 125 pupils for $=18$ days
Food will be sufficient for 1 pupil for $=18 \times 125$ days (less pupil more days)
and food will be sufficient for 150 pupils $=\frac{18 \times 125}{150}$ days(more pupil more days)
$=\frac{18 \times 5}{6}=15$ days
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