Let the base and altitude of the right-angled triangle be $x$ and $y \mathrm{~cm}$, respectively Therefore, the hypotenuse will be $(x+2) \mathrm{cm}$. $\therefore(x+2)^{2}=y^{2}+x^{2}$ Again, the...

Let the base be $x \mathrm{~m}$. Therefore, the altitude will be $(x+7) m$ $\begin{array}{l} \text { Area of a triangle }=\frac{1}{2} \times \text { Base } \times \text { Altitude } \\ \therefore...

Let the altitude of the triangle be $x \mathrm{~cm}$ Therefore, the base of the triangle will be $(x+10) \mathrm{cm}$ $\begin{array}{l} \text { Area of triangle }=\frac{1}{2} x(x+10)=600 \\...

Let the breadth of rectangle be $x \mathrm{~cm}$. According to the question: Side of the square $=(x+4) \mathrm{cm}$ Length of the rectangle $=\{3(x+4)\} \mathrm{cm}$ It is given that the areas of...

Let the width of the path be $x \mathrm{~m}$. $\therefore$ Length of the field including the path $=16+x+x=16+2 x$ Breadth of the field including the path $=10+x+x=10+2 x$ Now, (Area of the field...

Let the length and breadth of the rectangle be $2 x \mathrm{~m}$ and $x \mathrm{~m}$, respectively. According to the question: $\begin{array}{l} 2 x \times x=288 \\ \Rightarrow 2 x^{2}=288 \\...

Let the time taken by one pipe to fill the tank be $x$ minutes. $\therefore$ Time taken by the other pipe to fill the tank $=(x+5) \min$ Suppose the volume of the tank be $V$. Volume of the tank...

Let B takes $x$ days to complete the work. Therefore, A will take $(x-10)$ days. $\begin{array}{l} \therefore \frac{1}{x}+\frac{1}{(x-10)}=\frac{1}{12} \\ \Rightarrow...

Speed of the boat in still water $=8 \mathrm{~km} / \mathrm{hr}$. Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. $\therefore$ Speed upstream $=(8-x) \mathrm{km} / \mathrm{hr}$ Speed...

Let the speed of the Deccan Queen be $x \mathrm{~km} / \mathrm{hr}$. According to the question: Speed of another train $=(x-20) \mathrm{km} / \mathrm{hr}$ $\begin{array}{l} \therefore...

Let the usual speed $x \mathrm{~km} / \mathrm{hr}$. According to the question: $\begin{array}{l} \frac{300}{x}-\frac{300}{(x+5)}=2 \\ \Rightarrow \frac{300(x+5)-300 x}{x(x+5)}=2 \\ \Rightarrow...

Let the speed of the train be xkmph The time taken by the train to travel $180 \mathrm{~km}$ is $\frac{180}{\mathrm{x}} \mathrm{h}$ The increased speed is $\mathrm{x}+9$ The time taken is...

Let the shortest side be $x \mathrm{~m}$. Therefore, according to the question: Hypotenuse $=(2 x-1) m$ Third side $=(x+1) m$ On applying Pythagoras theorem, we get: $\begin{array}{l} (2...

Let the base and altitude of the right-angled triangle be $x$ and $y \mathrm{~cm}$, respectively Therefore, the hypotenuse will be $(x+2) \mathrm{cm}$. $\therefore(x+2)^{2}=y^{2}+x^{2}$ Again, the...

Let one side of the right-angled triangle be $x \mathrm{~m}$ and the other side be $(x+4) m$. On applying Pythagoras theorem, we have: $\begin{array}{l} 20^{2}=(x+4)^{2}+x^{2} \\ \Rightarrow...

Let the altitude of the triangle be $x \mathrm{~m}$. Therefore, the base will be $3 x \mathrm{~m}$. $\begin{array}{l} \text { Area of a triangle }=\frac{1}{2} \times \text { Base } \times \text {...

Let the length and breadth of the rectangular garden be $x$ and $y$ meter, respectively. Given: $x y=180 s q m.....(i)and$ $\begin{array}{l} 2 y+x=39 \\ \Rightarrow x=39-2 y \end{array}$ Putting the...

Let the length of the side of the first and the second square be $x$ and $y \cdot$ respectively. According to the question: $x^{2}+y^{2}=640$ Also, $\begin{array}{l} 4 x-4 y=64 \\ \Rightarrow x-y=16...

Let the length and breadth of the rectangular plot be $x$ and $y$ meter, respectively. Therefore, we have: $\begin{array}{l} \text { Perimeter }=2(x+y)=62 \quad \ldots . .(i) \text { and } \\ \text...

Let the length and breadth of the rectangle be $3 x \mathrm{~m}$ and $x \mathrm{~m}$, respectively. According to the question: $\begin{array}{l} 3 x \times x=147 \\ \Rightarrow 3 x^{2}=147 \\...

Let the tap of smaller diameter fill the tank in $x$ hours. $\therefore$ Time taken by the tap of larger diameter to fill the tank $=(x-9) h$ Suppose the volume of the tank be $V$. Volume of the...

Let one pipe fills the cistern in $x$ mins. Therefore, the other pipe will fill the cistern in $(x+3)$ mins. Time taken by both, running together, to fill the cistern $=3 \frac{1}{13} \min...

Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. $\therefore$ Downstream speed $=(9+x) \mathrm{km} / \mathrm{hr}$ Upstream speed $=(9-x) \mathrm{km} / \mathrm{hr}$ Distance covered...

Let the speed of the stream be $x \mathrm{~km} / \mathrm{hr}$. Given: Speed of the boat $=18 \mathrm{~km} / \mathrm{hr}$ $\therefore$ Speed downstream $=(18+x) \mathrm{km} / \mathrm{hr}$ Speed...

Let the usual speed $x \mathrm{~km} / \mathrm{hr}$. According to the question: $\begin{array}{l} \frac{300}{x}-\frac{300}{(x+5)}=2 \\ \Rightarrow \frac{300(x+5)-300 x}{x(x+5)}=2 \\ \Rightarrow...

Let the speed of the train be xkmph The time taken by the train to travel $180 \mathrm{~km}$ is $\frac{180}{\mathrm{x}} \mathrm{h}$ The increased speed is $\mathrm{x}+9$ The time taken is...

Let the first speed of the train be $x \mathrm{~km} / \mathrm{h}$. Time taken to cover $54 \mathrm{~km}=\frac{54}{x} h \quad\left(\right.$ Time $\left.=\frac{\text { Distance }}{\text { Speed...

Let the first speed of the truck be $x \mathrm{~km} / \mathrm{h}$. $\therefore$ Time taken to cover $150 \mathrm{~km}=\frac{150}{x} h \quad\left(\right.$ Time $\left.=\frac{\text { Dis tan ce...

Let son's age 2 years ago be $x$ years. Then, Man's age 2 years ago $=3 x^{2}$ years $\therefore$ Sons present age $=(x+2)$ years Man's present age $=\left(3 x^{2}+2\right)$ years In three years...

Let the present age of Meena be $x$ years. According to the question: $\begin{array}{l} (x-5)(x+8)=30 \\ \Rightarrow x^{2}+3 x-40=30 \\ \Rightarrow x^{2}+3 x-70=0 \\ \Rightarrow x^{2}+(10-7) x-70=0...

Let the present age of Meena be $x$ years Meena's age 3 years ago $=(x-3)$ years Meena's age 5 years hence $=(x+5)$ years According to the given condition, $\frac{1}{x-3}+\frac{1}{x+5}=\frac{1}{3}$...

Let the cost price of the article be $x$ $\therefore$ Gain percent $=x \%$ According to the given condition, $\Rightarrow \frac{100 x+x^{2}}{100}=75$ $\Rightarrow x^{2}+100 x=7500$ $\Rightarrow...

Let the marks obtained by $P$ in mathematics and science be $x$ and $(28-x)$, respectively. According to the given condition, $\begin{array}{l} (x+3)(28-x-4)=180 \\ \Rightarrow(x+3)(24-x)=180 \\...

Let the original price of the book be $Rs x$. $\therefore$ Number of books bought at original price for $Rs. 600=\frac{600}{x}$ $\therefore$ Number of books bought at reduced price for...

Let $x$ be the number of students who planned a picnic. $\therefore$ Original cost of food for each member $=$ γ $\frac{2000}{x}$ Five students failed to attend the picnic. So, $(x-5)$ students...

Let the marks of Kamal in mathematics and English be $x$ and $y$, respectively. According to the question: $x+y=40$ Also, $\begin{array}{l} (x+3)(y-4)=360 \\ \Rightarrow(x+3)(40-x-4)=360 \quad[\text...

Let the required number be $x$. According to the given condition, $\begin{array}{l} x+\frac{1}{x}=2 \frac{1}{30} \\ \Rightarrow \frac{x^{2}+1}{x}=\frac{61}{30} \\ \Rightarrow 30 x^{2}+30=61 x \\...

Let the numerator be $x$. $\therefore$ Denominator $=x+3$ $\therefore$ Original number $=\frac{x}{x+3}$ According to the question: $\frac{x}{x+3}+\frac{1}{\left(\frac{x}{x+3}\right)}=2 \frac{9}{10}$...

Let the digits at units and tens places be $x$ and $y$, respectively. Original number $=10 y+x$ According to the question: $\begin{array}{l} 10 y+x=4(x+y) \\ \Rightarrow 10 y+x=4 x+4 y \\...

Let the greater number be $x$ and the smaller number be $y$. According to the question: $\begin{array}{l} x^{2}-y^{2}=45 \\ y^{2}=4 x \end{array}$ From (i) and (ii), we get: $x^{2}-4 x=45$...

Let the larger and smaller parts be $x$ and $y$, respectively. According to the question: $\begin{array}{l} x+y=16 .....(i) \\ 2 x^{2}=y^{2}+164.....(ii) \end{array} \quad$ From (i), we get:...

Let the two parts be $x$ and $(57-x)$. According to the given condition, $\begin{array}{l} x(57-x)=680 \\ \Rightarrow 57 x-x^{2}=680 \\ \Rightarrow x^{2}-57 x+680=0 \\ \Rightarrow x^{2}-40 x-17...

Let the required consecutive multiplies of 7 be $7 x$ and $7(x+1)$. According to the given condition, $(7 x)^{2}+[7(x+1)]^{2}=1225$ $\Rightarrow 49 x^{2}+49\left(x^{2}+2 x+1\right)=1225$...

Let the required natural numbers be $x$ and $(x+3)$. Now, $x<x+3$ $\therefore \frac{1}{x}>\frac{1}{x+3}$ According to the given condition, $\begin{array}{l}...

Let the required natural numbers be $x$ and $(9,-x)$. According to the given condition, $\begin{array}{l} \frac{1}{x}+\frac{1}{9-x}=\frac{1}{2} \\ \Rightarrow \frac{9-x+x}{x(9-x)}=\frac{1}{2} \\...

Let the two consecutive positive odd integers be $x$ and $(x+2)$. According to the given condition, $\begin{array}{l} x(x+2)=483 \\ \Rightarrow x^{2}+2 x-483=0 \\ \Rightarrow x^{2}+23 x-21 x-483=0...

Let the required numbers be $x$ and $(x+3)$. According to the question: $\begin{array}{l} x(x+3)=504 \\ \Rightarrow x^{2}+3 x=504 \\ \Rightarrow x^{2}+3 x-504=0 \\ \Rightarrow x^{2}+(24-21) x-504=0...

Let the two consecutive positive even numbers be $x$ and $(x+2)$. According to the given condition, $\begin{array}{l} x^{2}+(x+2)^{2}=452 \\ \Rightarrow x^{2}+x^{2}+4 x+4=452 \\ \Rightarrow 2...

Let the required two consecutive positive integers be $x$ and $(x+1)$. According to the given condition, $\begin{array}{l} x^{2}+(x+1)^{2}=365 \\ \Rightarrow x^{2}+x^{2}+2 x+1=365 \\ \Rightarrow 2...