Application of Linear Equation

**Category : **7th Class

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)

- An equation is a statement which contains one or more than one variables.
- An equation in which the highest power of variable is one is called linear equation.
- The value of variable which satisfies the given linear equation is called solution.
- Generally there are three methods to solve a linear equation.

(a) Trial and error method.

(b) Systematic method.

(c) Transposition method.

- Do you know a linear equations appear with great regularity because so many measurable quantities are proportional to other quantities as in related linearly.
- Linear equations are helpful first approximations of computationally prohibitive nonlinear phenomena.

*play_arrow*Linear Equation*play_arrow*Application of Linear Equation*play_arrow*Notes - Linear Equation in one variable

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