[4 marks sum]

Question 14 Marks

If $1 \frac{1}{4}: 2 \frac{1}{3}=p: q$ and $q: r=4 \frac{1}{2}: 5 \frac{1}{4} ;$ find $p: r$.

Answer

$\begin{aligned} & \mathrm{p}: \mathrm{q}=1 \frac{1}{4}: 2 \frac{1}{3}=\frac{5}{4}: \frac{7}{3} \\ & \frac{\mathrm{p}}{\mathrm{q}}=\frac{5}{4} \times \frac{3}{7}=\frac{15}{28} \\ & \mathrm{q}: \mathrm{r}=4 \frac{1}{2}: 5 \frac{1}{4}=\frac{9}{2}: \frac{21}{4} \\ & \frac{\mathrm{q}}{\mathrm{r}}=\frac{9}{2} \times \frac{4}{21}=\frac{6}{7} \\ & \therefore \frac{p}{q} \times \frac{q}{r}=\frac{15}{28} \times \frac{6}{7} \\ & \Rightarrow \frac{\mathrm{p}}{\mathrm{r}}=\frac{45}{98} \\ & \therefore \mathrm{p}: \mathrm{r}=45: 98\end{aligned}$

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Question 24 Marks

Find the fourth proportional of 4 hours 40 minutes, 1 hour 10 minutes and 16 hours.

Answer

Fourth proportional of 4 hours 40 minutes, 1 hour 10 minutes and 16 hours.
4 hours 40 minutes $=4 \times 60+40$
$ =24040=280 $
1 hour 10 minutes $=1 \times 60+10$
$ =60+10=70 \text { minutes } $
16 hours $=16 \times 60=960$ minutes
$\therefore$ Fourth proportional $=\frac{70 \times 960}{280}$
$=240$ minutes $=\frac{240}{60}=4$ hours

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Question 34 Marks

Two numbers are in the ratio 4 : 7. If their L.C.M. is 168, find the numbers.

Answer

Given, Ratio in two numbers $=4: 7$
and their L.C.M. $=168$
Let first number $=4 x$
and second number $=7 x$
Now, L.C.M. of $4 x$ and $7 x$
$ =4 \times 7 \times x=28 x $
$\therefore 28 x=168$
$ x=\frac{168}{28} $
$x=6$
$\therefore$ Required numbers $=4 x$ and $7 x=4 \times 6=24$ and $7 \times 6=42$

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Question 44 Marks

A plot of land, 600 sq m in area, is divided between two persons such that the first person gets three-fifth of what the second gets. Find the share of each.

Answer

Area of plot of land $=600$ sq. meter
Let second's share $=\mathrm{x}$
Then first share $=\frac{3}{5} x$
$\therefore$ Ratio between them $\frac{3}{5} x: x$
$ =\frac{3}{5}: 1=3: 5 $
Sum of ratios $=3+5=8$
$\therefore$ Share of first person $=\frac{600}{8} \times 3=225$ sq.m
and second share $=\frac{600}{8} \times 5=375$ sq.m.

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Question 54 Marks

Three persons start a business and spend Rs. 25,000; Rs. 15,000 atid Rs. 40,000 respectively. Find the share of each out of a profit of Rs. 14,400 in a year

Answer

$A^{\prime}$ 's investment $=$ Rs. 25000
B's investment $=$ Rs. 15000
C's investment $=$ Rs. 40000
$\therefore$ Ratio between their investment
$=25000: 15000: 40000$
$=5: 3: 8$
Sum of ratios $=5+3+8=16$ Total profit ₹ 14400
$\therefore$ A's share $=\frac{14400}{16} \times 5=$ ₹4500
B's share$=\frac{14400}{16} \times 3=$ ₹ 2700
C's share $=\frac{14400}{16} \times 8=$ ₹ 7200

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Question 64 Marks

The ages of two boys A and B are 6 years 8 months and 7 years 4 months respectively. Divide Rs. 3,150 in the ratio of their ages.

Answer

$A^{\prime}$ s age $=6$ years 8 months
$ =6 \times 12+8=72+8=80 \text { months } $
B's age $=7$ years 4 months $=7 \times 12+4=84+4=88$ months
$\therefore$ Ratio between them $=80: 88=10: 11$
Amount $=$ Rs. 3150
Sum of ratios $=10+11=21$
$\therefore$ A's share $=\frac{3150 \times 10}{21}=1500=$ Rs. 1500
B's share $=\frac{3150 \times 11}{21}=1650=$ Rs. 1650

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Question 74 Marks

The sum of three numbers, whose ratios are $3 \frac{1}{3}: 4 \frac{1}{5}: 6 \frac{1}{8}$ is 4917 . Find the numbers.

Answer

Sum of three numbers $=4917$
Ratio between them $=3 \frac{1}{3}: 4 \frac{1}{5}: 6 \frac{1}{8}$
$ =3 \frac{1}{3}: 4 \frac{1}{5}: 6 \frac{1}{8}=\frac{10}{3}: \frac{21}{5}: \frac{49}{8} $
$=\mathrm{LCM}$ of denominators 3, 5, and 8 is 120.
$ =\frac{10.120}{3}: \frac{21.120}{5}: \frac{49.120}{8} $
$ =10.40: 21.24: 49.15 $
Multiplication of the above Numbers, we will get this,
$ =400: 504: 735 $
Sum of ratio's $=400+504+735=1639$
$\therefore$ First number $=\frac{400}{1639}$ of $4917=1200$
Second number $=\frac{504}{1639}$ of $4917=1512$
and third number $=\frac{735}{1639}$ of $4917=2205$
Therefore, the first, second, and third number is 1200, 1512, and 2205 respectively.

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Question 84 Marks

A bag contains ₹ 1,600 in the form of ₹ 10 and ₹ 20 notes. If the ratio between the numbers of ₹ 10 and ₹ 20 notes is 2 : 3; find the total number of notes in all.

Answer

Total amount in the bag $=1600$
It contains notes in the denomination of ₹ 10 and 20
Ratio between the number of ₹ 10 and 20 notes is $=2: 3$
Let number of ₹ 10 note $=x$
and number of ₹ 20 notes $=y$
According to condition,
$ 10 x+20 y=1600..........(i) $
and $\mathrm{x}=\frac{2}{3} \mathrm{y}$..........(Ii)
Now, substitute the value of $x$ in eq (i)
$ 10 \times \frac{2}{3} y+20 y=1600 $
$\Rightarrow \frac{20}{3} \mathrm{y}+20 y=1600$
$\Rightarrow \frac{20+60}{3} y=1600$
$\Rightarrow \frac{80}{3} \mathrm{y}=1600$
$\Rightarrow \mathrm{y}=\frac{1600 \times 3}{80}$
$\therefore y=60$
Now, substitute the value of $y$ in eq (ii), we get
$ x=\frac{2}{3} \times 60=40 $
Total nuber of notes in all $=x+y$
$ =60+40=100 \text { notes } $

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MATHS [4 marks sum]

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