The area of a trapezium is 475 cm2 and the height is 19 cm.
Question: The area of a trapezium is 475 cm2 and the height is 19 cm. Find the lengths of its two parallel sides, if one side is 4 cm greater than the other. Solution: Let the side of a trapezium, $D C=x \mathrm{~cm}$ According to the question, Other side, $A B=(x+4) \mathrm{cm}$ We know that, Area of a trapezium $=\frac{1}{2}$ (Sum of parallel sides) $\times$ Distance between parallel sides [ $\because$ area of a trapezium $=\frac{1}{2}(a+b) \times h$ ] $\Rightarrow \quad 475=\frac{1}{2}(x+x+4)...
Read More →A metallic bucket, open at the top, of height 24 cm
Question: A metallic bucket, open at the top, of height 24 cm is in the form of thefrustum of a cone, the radii of whose lower and upper circular ends are7 cm and 14 cm, respectively. Find(i) the volume of water which can completely fill the bucket;(ii) the area of the metal sheet used to make the bucket. Solution: We have, Height, $h=24 \mathrm{~cm}$, Upper base radius, $R=14 \mathrm{~cm}$ and lower base radius, $r=7 \mathrm{~cm}$ Also, the slant height, $l=\sqrt{(R-r)^{2}+h^{2}}$ $=\sqrt{(14-7...
Read More →Some toffees are bought at the rate of 11 for Rs 10 and the same number at the rate of 9 for Rs 10.
Question: Some toffees are bought at the rate of 11 for Rs 10 and the same number at the rate of 9 for Rs 10. If the whole lot is sold at one rupee per toffee, find the gain or loss percent on the whole transaction. Solution: Let the total number of toffees bought be $x$. Let $\frac{x}{2}$ toffees at the rate of 11 are bought for Rs. 10, and $\frac{x}{2}$ toffees at the rate of 9 are bought for Rs. 10 Total money spent on buying the toffees $=\left(\frac{x}{2}\right)\left(\frac{10}{11}\right)+\l...
Read More →The perimeter of a triangle is 50 cm.
Question: The perimeter of a triangle is 50 cm. One side of a triangle is 4 cm longer than the smaller side and the third side is 6 cm less than twice the smaller side. Find the area of the triangle. Solution: Let the smaller side of a triangle be $x \mathrm{~cm}$. According to the question, One side $=4 \mathrm{~cm}$ longer than the smaller side $=(x+4) \mathrm{cm}$ Third side $=6 \mathrm{~cm}$ less than twice the smaller side $=(2 x-6) \mathrm{cm}$ $\therefore$ Perimeter of a triangle $=50 \ma...
Read More →The radii of the circular ends of a solid frustum of a cone are 18 cm and 12 cm
Question: The radii of the circular ends of a solid frustum of a cone are 18 cm and12 cm and its height is 8 cm. Find its total surface area.[Use= 3.14] Solution: We have, Height, $h=8 \mathrm{~cm}$, Base radii, $R=18 \mathrm{~cm}$ and $r=12 \mathrm{~cm}$ Also, the slant height, $l=\sqrt{(R-r)^{2}+h^{2}}$ $=\sqrt{(18-12)^{2}+8^{2}}$ $=\sqrt{6^{2}+8^{2}}$ $=\sqrt{36+64}$ $=\sqrt{100}$ $=10 \mathrm{~cm}$ Now, Total surface area of the solid frustum $=\pi(R+r) l+\pi R^{2}+\pi r^{2}$ $=3.14 \times(1...
Read More →Mariam bought two fans for Rs 3605.
Question: Mariam bought two fans for Rs 3605. She sold one at a profit of 15% and the other at a loss of 9%. If Mariam obtained the same amount for each fan, find the cost price of each fan. Solution: It is given that the $S$. $P$ is same for both the fans. Let C.P of the first fa $n$ be Rs. $\mathrm{x}$ Therefore, C.P of the second fan $=$ Rs. $(3605-\mathrm{x})$ Profit on the first fan $=15 \%$ Loss on the second fan $=9 \%$ For the first fan, S.P $=$ C.P $\left(\frac{100+\text { gain } \%}{10...
Read More →How much paper of each shade is needed
Question: How much paper of each shade is needed to make a kite given in figure, in which ABCD is a square with diagonal 44 cm. Solution: We know that, all the sides of a square are always equal. i.e., $\quad A B=B C=C D=D A$ In $\triangle A C D$ $A C=44 \mathrm{~cm}, \angle D=90^{\circ}$ Using Pythagoras theorem in $\triangle A C D$, $A C^{2}=A D^{2}+D C^{2}$ $\Rightarrow$ $44^{2}=A D^{2}+A D^{2}$ $[\because D C=A D]$ $\Rightarrow$ $2 A D^{2}=44 \times 44$ $\Rightarrow$ $A D^{2}=22 \times 44 \R...
Read More →A drinking glass is in the shape of a frustum of a cone of height 14 cm.
Question: A drinking glass is in the shape of a frustum of a cone of height 14 cm. The diameters of its two circular ends are 16 cm and 12 cm. Find the capacity of the glass. Solution: We have, Height of the frustum, $h=14 \mathrm{~cm}$, Base radii, $R=\frac{16}{2}=8 \mathrm{~cm}$ and $r=\frac{12}{2}=6 \mathrm{~cm}$ The capacity of the glass = Volume of the frustum $=\frac{1}{3} \pi h\left(R^{2}+r^{2}+r R\right)$ $=\frac{1}{3} \times \frac{22}{7} \times 14 \times\left(8^{2}+6^{2}+8 \times 6\righ...
Read More →In a hospital, used water is collected in a cylindrical tank of diameter 2 m and height 5 m.
Question: In a hospital, used water is collected in a cylindrical tank of diameter 2 m and height 5 m. After recycling, this water is used to irrigate a park of hospital whose length is 25 m and breadth is 20 m. If the tank is filled completely then what will be the height of standing water used for irrigating the park? Write your views on recycling of water. Solution: Diameter of cylinder (d) = 2 mRadius of cylinder (r) = 1 mHeight of cylinder (H) = 5 m Volume of cylindrical tank, $\mathrm{V}_{...
Read More →A dealer bought two tables for Rs 3120.
Question: A dealer bought two tables for Rs 3120. He sold one of them at a loss of 15% and other at a gain of 36%. Then, he found that each table was sold for the same price. Find the cost price of each table. Solution: Los $s$ on the first table $=15 \%$ Therefore, $S . P=C . P\left(\frac{100-l \text { oss } \%}{100}\right)$ S. P $=\frac{85 x}{100}=$ Rs. $0.85 x$ Gain on the second table $=36 \%$ Therefore, S.P $=C . P\left(\frac{100+g \text { ain } \%}{100}\right)$ $S . P=$ Rs. $1.36(3120-x)$ ...
Read More →A dishonest shopkeeper professes to sell pulses at his cost price but uses a false weight of 950 gm for each kilogram.
Question: A dishonest shopkeeper professes to sell pulses at his cost price but uses a false weight of 950 gm for each kilogram. Find his gain percent. Solution: He sells $950 \mathrm{gm}$ pulses and gets a gain of $50 \mathrm{gm}$. If he sells $100 \mathrm{gm}$ of pulses, he will gain $=\frac{50}{950} \times 100$ $=\frac{5000}{950}$ $=5 \frac{5}{19} \%$...
Read More →A man sells an article at a profit of 25%.
Question: A man sells an article at a profit of 25%. If he had bought it at 20% less and sold it for Rs 36.75 less, he would have gained 30%. Find the cost price of the article. Solution: Let the C. P of the article be Rs. $x$. Original S.P $=x+\frac{25}{100} x$ $=R s . \frac{5 x}{4}$ If he purchased it at $20 \%$ less, $C . \mathrm{P}=x-\frac{20}{100} x$ $=R s . \frac{4 x}{5}$ He sold the article at Rs $36.75$ less. So, the selling price $=R s . \frac{5 \mathrm{x}}{4}-36.75$ Given that he would...
Read More →Find the area of the trapezium PQRS
Question: Find the area of the trapezium PQRS with height PQ given in the figure given below Solution: We have, trapezium $P Q R S$, in which draw a line $R T$ perpendicular to $P S$. where, side, $S T=P S-T P=12-7=5 \mathrm{~m} .$ $[\because T P=P Q=7 \mathrm{~m}]$ $\begin{array}{lll}\text { In right angled } \Delta S T R, (S R)^{2}=(S T)^{2}+(T R)^{2} \text { [by using Pythagoras theorem] }\end{array}$ $\Rightarrow \quad(13)^{2}=(5)^{2}+(T R)^{2}$ $\Rightarrow \quad(T R)^{2}=169-25$ $\Rightarr...
Read More →A right triangle whose sides are 15 cm and 20 cm (other than hypotenuse), is made to revolve about its hypotenuse.
Question: A right triangle whose sides are 15 cm and 20 cm (other than hypotenuse), is made to revolve about its hypotenuse. Find the volume and surface area of the double cone so formed. (Choose value ofas found appropriate) Solution: We have, In $\Delta \mathrm{ABC}, \angle \mathrm{B}=90^{\circ}, \mathrm{AB}=l_{1}=15 \mathrm{~cm}$ and $\mathrm{BC}=l_{2}=20 \mathrm{~cm}$ Let $\mathrm{OD}=\mathrm{OB}=r, \mathrm{AO}=h_{1}$ and $\mathrm{CO}=h_{2}$ Using Pythagoras theorem, $\mathrm{AC}=\sqrt{\math...
Read More →A rhombus shaped sheet with perimeter 40 cm
Question: A rhombus shaped sheet with perimeter 40 cm and one diagonal 12 cm, is painted on both sides at the rate of Rs. 5 per cm2. Find the cost of painting. Solution: Let $A B C D$ be a rhombus having each side equal to $x \mathrm{~cm}$. i.e., $\quad A B=B C=C D=D A=x \mathrm{~cm}$ Given, perimeter of a rhombus $=40$ $\therefore \quad A B+B C+C D+D A=40$ $\Rightarrow \quad x+x+x+x=40$ $\Rightarrow \quad 4 x=40$ $\Rightarrow \quad x=\frac{40}{4}$ $x=10 \mathrm{~cm}$ In $\triangle A B C$, let $...
Read More →By selling 90 ball pens for Rs 160 a person loses 20%.
Question: By selling 90 ball pens for Rs 160 a person loses 20%. How many ball pens should be sold for Rs 96 so as to have a profit of 20%? Solution: S. $P$ of 90 ball pens $=$ Rs. 160 Loss $=20 \%$ Therefore, C. $\mathrm{P}=\operatorname{SP}\left(\frac{100}{100-l \text { oss } \%}\right)$ $\mathrm{CP}=\frac{100}{100-20} \times 160$ $=\frac{16000}{80}$ $=$ Rs. 200 Now, S.P of 90 ball pens $=$ Rs. 96 Profit $=20 \%$ C.P $=\mathrm{SP}\left(\frac{100}{100+\text { Profit\% }}\right)$ $\mathrm{CP}=\f...
Read More →A defective briefcase costing Rs 800 is being sold at a loss of 8%.
Question: A defective briefcase costing Rs 800 is being sold at a loss of 8%. If the price is further reduced by 5%, find its selling price. Solution: C. $\mathrm{P}$ of the briefcase $=$ Rs. 800 Loss $=8 \%$ S. P $=$ CP $\left(\frac{100-\text { loss } \%}{100}\right)$ $=800\left(\frac{100-8}{100}\right)$ $=800 \times 0.92=$ Rs. 736 If the $p$ rice is further reduced by $5 \%$, the $s$ elling price of the briefcase will be $=$ Rs. $\left(736-736 \times \frac{5}{100}\right)$ $=736\left(\frac{100-...
Read More →By selling a book for Rs 258, a bookseller gains 20%.
Question: By selling a book for Rs 258, a bookseller gains 20%. For how much should he sell it to gain 30%? Solution: Selling price of the book $=$ Rs. 258 Gain $\%=20 \%$ Since, C.P $=\left[\frac{100}{(100+\text { Gain } \%)} \times S . P\right]$ $=\left[\frac{100}{(100+20)} \times 258\right]$ $=\frac{25800}{120}$ $=$ Rs. 215 Let the bookseller sells it for Rs. $x$ So, $S . P=\left[\frac{(100+\text { Gain } \%)}{100} \times C . P\right]$ $x=\frac{100+30}{100} \times 215$ $=\frac{130 \times 215}...
Read More →Ravish sold his motorcycle to Vineet at a loss of 28%.
Question: Ravish sold his motorcycle to Vineet at a loss of 28%. Vineet spent Rs 1680 on its repairs and sold the motor cycle to Rahul for Rs 35910, thereby making a profit of 12.5%, find the cost price of the motor cycle for Ravish. Solution: Let the cost price of the motorcycle for Ravis $h$ be Rs. $\mathrm{x}$. $\operatorname{Loss} \%=28 \%$ Therefore, $\mathrm{SP}=\mathrm{CP}\left(\frac{100-\text { Loss } \%}{100}\right)$ $\mathrm{SP}=$ Rs. $x\left(\frac{72}{100}\right)$ Selling price of the...
Read More →If the selling price of 18 oranges is equal to the cost price of 16 oranges,
Question: If the selling price of 18 oranges is equal to the cost price of 16 oranges, find the loss percent. Solution: Let the cost price of one orange be Rs. C, and its selling price be Rs. S Therefore, $16 \mathrm{C}=18 \mathrm{~S}$ $\mathrm{C}=\frac{18}{16} \mathrm{~S}$ As cost price is more than the selling price, S. P. $=\left(\frac{100-\text { loss } \%}{100}\right)$ C. P $\mathrm{S}=\left(\frac{100-\text { loss } \%}{100}\right) \mathrm{C}$ $\frac{\mathrm{S}}{\mathrm{C}}=\left(\frac{100-...
Read More →If the cost price of 18 chairs be equal to selling price of 16 chairs,
Question: If the cost price of 18 chairs be equal to selling price of 16 chairs, find the gain or loss percent. Solution: Let the cost price of one chair be Rs. C, and selling price be Rs. S There fore, $18 \mathrm{C}=16 \mathrm{~S}$ $\mathrm{C}=\frac{16}{18} \mathrm{~S}$ As cost price is less than the selling price, S. P. $=\left(\frac{100+\text { profit } \%}{100}\right)$ C.P $\mathrm{S}=\left(\frac{100+\text { profit } \%}{100}\right) \mathrm{C}$ $\frac{\mathrm{S}}{\mathrm{C}}=\left(\frac{100...
Read More →If the selling price of 10 pens is equal to cost price of 14 pens,
Question: If the selling price of 10 pens is equal to cost price of 14 pens, find the gain percent. Solution: Let the cost price of one pen be Rs. C, and the selling price be Rs. S Therefore, $10 \mathrm{~S}=14 \mathrm{C}$ $\mathrm{C}=\frac{10}{14} \mathrm{~S}$ However, the cost price is less than the selling price. S. P. $=\left(\frac{100+\text { profit } \%}{100}\right)$ C.P S $=\left(\frac{100+\text { profit } \%}{100}\right)$ C $\frac{\text { S }}{\text { C }}=\left(\frac{100+\text { profit ...
Read More →The sides of a quadrilateral ABCD are 6 cm,
Question: The sides of a quadrilateral ABCD are 6 cm, 8 cm, 12 cm and 14 cm (taken in order), respectively and the angle between the first two sides is a right angle. Find its area. Solution: Given ABCD is a quadrilateral having sides AB = 6 cm, BC = 8 cm, CD = 12 cm and DA = 14 cm. Now, join AC. We have, $A B C$ is a right angled triangle at $B$. Now, $\quad A C^{2}=A B^{2}+B C^{2} \quad$ [by Pythagoras theorem] $=6^{2}+8^{2}=36+64=100$ $\Rightarrow \quad A C=10 \mathrm{~cm} \quad$ [taking posi...
Read More →A copper wire of diameter 6 mm is evenly wrapped on a cylinder of length 18 cm
Question: A copper wire of diameter 6 mm is evenly wrapped on a cylinder of length 18 cm and diameter 49 cm to cover its whole surface. Find the length and the volume of the wire. If the density of the copper be 8.8 g per cm3, then find the weight of the wire. Solution: We have, Diameter of the coppe wire, $d=6 \mathrm{~mm}=0.6 \mathrm{~cm}$, Radius of the copper wire, $r=\frac{0.6}{2}=0.3 \mathrm{~cm}$, Length of the cylinder, $H=18 \mathrm{~cm}$, Radius of the cylinder, $R=\frac{49}{2} \mathrm...
Read More →Ramesh bought two boxes for Rs 1300.
Question: Ramesh bought two boxes for Rs 1300. He sold one box at a profit of 20% and the other box at a loss of 12%. If the selling price of both boxes is the same, find the cost price of each box. Solution: Let the cost price of the first box be Rs. $x$. Therefore, the $\cos t$ of the second box will be Rs. $(1300-x)$ Profit on the first box $=20 \%$ Loss on the second box $=12 \%$ $\mathrm{SP}$ of the first box $=\mathrm{CP}\left(\frac{\text { gain } \%+100}{100}\right)$ $\mathrm{SP}=\mathrm{...
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