Application of Linear Equation
Category : 7th Class
Application of Linear Equation
When you are solving the word problem you should follow the following steps:
Step 1: Read the problem carefully and specify the given and required parameters.
Step 2: Represent the unknown quantity by variables like x, y w....etc.
Step 3: Convert the mathematical statements into mathematical problem.
Step 4: Use the conditions to form an equation.
Step 5: Solve the equation for the unknown and check whether the solution satisfies the equating or not.
The sum of three consecutive multiples of 8 is 888. Which one of following options is the group of those numbers?
(a) 504, 342 and 342
(b) 234, 567 and 604
(c) 234, 564 and 905
(d) 288, 296 and 304
(e) None of these
Answer: (d)
Explanation
Let the first multiple of 8 be 8x then the next two multiples of 8 will be \[8(x+1)\And 8(x+2)\]
It is given that the sum of these three consecutive multiples is 888.
\[\therefore 8x+8(x+1)+8(x+2)\text{ }=888\]
\[\Rightarrow 8x+8x+8+8x+16\]\[=888\text{ }\Rightarrow 24x+24=888\]
\[\Rightarrow 24x=888-24\Rightarrow 24x=864\Rightarrow x=\frac{864}{24}=36\]
Therefore, three consecutive multiples of 8 are, \[8\times 36,8\times 37\And 8\times 38.\]i.e., 288, 296 and 304
The denominator of a rational number is greater than its numerator by 6. If enumerator is increased by 5 and the denominator is decreased by 3 then the number obtained is \[\frac{5}{4},\] find the rational number.
(a) \[\frac{5}{11}\]
(b) \[\frac{11}{5}\]
(c) \[\frac{12}{3}\]
(d) \[\frac{9}{8}\]
(e) None of these
Answer: (a)
Explanation
Let the numerator of the rational number be x.
Then the denominator of the rational number will be \[x+6\]
It is given that the numerator and denominator of the number are increased and decreased by 5 and 3 respectively then the number obtained is \[\frac{5}{4}\]
\[\therefore \] Numerator of the new rational number\[~=x\text{+}5\]
Denominator of the new rational number \[=(x+6)-3=x+3\]
\[\therefore \] New rational number \[=\frac{x+5}{x+3}\]
But the new rational number is given as \[\frac{5}{4}\]
\[~\therefore \frac{x+5}{x+3}=\frac{5}{4}\Rightarrow 4(x+5)=5(x+3)\](By cross multiplication)
\[\Rightarrow 4x+20=5x+15\] \[4x-5x=15-20\][transposing 5x to L.H.S. and 20 to R.H.S.]
\[\Rightarrow -x=-5\Rightarrow \] or \[x=5\]
\[\therefore \] Numerator of the rational number \[=5\]
Denominator of the rational number \[=5+6=11\]
\[\therefore \] The required rational number \[=\frac{5}{11}\]
A steamer goes downstream from one port to another in 6 hours. It covers the same distance up stream in 7 hours. If the speed of the stream is 2 km/hours then find the speed of the steamer in still water.
(a) 20km/h
(b) 30 Km/h
(c) 26 Km/h
(d) 48Km/h
(e) None of these
Answer: (c)
Explanation
Let the speed of the steamer in still water be \[x\text{ }Km/h\]
It is given that while going down stream the steamer takes 6 hours to cover the distance between two ports.
\[\therefore \] Speed of the steamer down stream \[=(x+2)\] Km/h.
Distance covered in \[1\text{ }h=(x+2)Km\]
Distance covered in \[6h=6(x+2)\text{ }Km\]
\[\therefore \] Distance between\[2\text{ }ports=6(x+2)\text{ }Km\] ............... (i)
It is given that while going up stream, the steamer takes 7 hours to cover the distance.
Speed of the steamer up stream \[=(x-2)\text{ }Km/h\]
Distance covered in \[1h=(x-2)\text{ }Km\]
Distance covered in \[7h=7(x-2)\text{ }Km\]
\[\therefore \]Distance covered in this case\[=7\text{(}x-2\text{)}km\]............(ii)
The distance between two ports is same
\[\therefore \]From (i) & (ii) we get
\[6(x+2)=7(x-2)\] \[6x+12=7x-14\]
\[\Rightarrow 6x-7x=-14-12\][Transposing \[7x\] to L.H.S. & 12 to R.H.S.]
\[\Rightarrow -x=-26\Rightarrow x=26\]
\[\therefore \] The speed of the streamer in still water \[=26Km/hrs\]
The present ages of Peter & Jony are in the ratio of 4 : 3, four years late their ages will be in the ratio of 6 : 5. What is their present ages?
(a) 8 years and 9 years
(b) 6 years and 9 years
(c) 8 years and 6 years
(d) 5 years and 9 years
(e) None of these
Answer: (c)
Explanation
Since the ratio of the present ages of Peter & Jony is given as \[4:3.\]
Let the present age of Peter is \[3x\] years, and present age of Jony is \[4x\] years
After four years
Peter's age \[=(4x+4)\] years
Jony's age \[=(3x+4)\] years
According to the given condition
\[(4x+4):(3x+4)=6:5\]
\[\frac{4x+4}{3x+4}=\frac{6}{5}\Rightarrow 5(4x+4)=\]\[6(3x+4)\Rightarrow 20x+20=18x+24\]
\[\Rightarrow 20x-18x=24-20\][Transposing \[18x\] to L.H.S. & 20 to R.H.S.]
\[\Rightarrow 2x=4,\]or \[x=2\]
\[\therefore \]Present age of Peter \[=(4x2)\] years, i.e. 8 years.
\[\therefore \]Present age of Jony \[=(3x2)\] years, i.e. 6 years.
The sum of the digits of a two digit numbers is 10. The number obtain by interchanging the digits exceeds the original number by 54, find original number.
(a) 29
(b) 28
(c) 55
(d) 95
(e) None of these
Answer: (b)
Explanation
Since the required number is a two digit number so, we have to find its units digit & tens digit.
Let the digit at ones place be\[~x.\]
It is given that the sum of the digit of the number is 10.
\[\therefore \]The digit at the tens place \[=10-x\]
Thus the original number \[=\text{ }10x(10-x)+x\]
\[=100\text{ -10}x+x\] \[=100-9x\]
On interchanging the digits of the given number the digit at the ones place becomes\[~(10-x)\] & the digit at the tens place becomes\[x.\]
\[\therefore \] New number \[=10x+(10-x)=9x+10\]
It is given that the new number exceeds the original number by 54.
i.e. New number-original number \[=54\] \[(9x+10)-(100-9x)=54\]
\[\Rightarrow 9x+10-100+9x=54\]Or, \[18x-90=54\]
\[\Rightarrow 18x=54+90\]or, \[18x=144\] or, \[x=\frac{144}{18}=8\]
\[\therefore \] The digit at the ones place\[~=8\]
The digit at the tens place \[=(10-8)=2\]
\[\therefore \] Original number \[=28\]
A Monkey climbing up a pole ascends 10 meters and slep down 2 metres in alternate minutes. If the pole is 57 metres high, how long will take him to reach the top of pole?
(a) 14 minutes, 6 seconds
(b) 16 minutes, 4 seconds
(c) 20 minutes, 30 seconds
(d) 10 minutes, 18 seconds
(e) None of these
Answer: (a)
Two trains of equal length are running on parallel tracks in the same direction at 46 km per hour. The faster train passes the slower train in 36 /seconds, the length of each train is:
(a) 46m
(b) 33m
(c) 53m
(d) cannot be determined
(e) None of these
Answer: (d)
The denominator of a number is greater than its numerator by 8. If the numerator increased by one the number obtained is\[\frac{2}{3}.\] The number is:
(a) \[\frac{3}{11}\]
(b) \[\frac{13}{21}\]
(c) \[\frac{11}{19}\]
(d) \[\frac{14}{22}\]
(e) None of these
Answer: (b)
(a) Trial and error method.
(b) Systematic method.
(c) Transposition method.
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