(i) The area of a rectangular plot is 528m^2. The length of the plot (in metres) is one more than twice its breadth. We need to find the length and breadth of the plot.
(ii) The product of two consecutive positive integers is 306. We need to find the integers.
(iii) Rohan's mother is 26 years older than him. The product of their ages (in years) 3 years from now will be 360. We would like to find Rohan's present age.
(iv) A train travels a distance of 480 km at a uniform speed. If the speed had been 8 km/h less, then it would have taken 3 hours more to cover the same distance. We need to find the speed of the train.
Answers
Answer (i)
Area = 528 m²
Let Breadth = x m
So, Length = (2x+1)
We use Use Area of Rectangle = Length × Breadth
Therefore,
➥ 528 = (x) × (2x+1)
➥ 528 = 2x² - x
➥ 2x² +x - 528
Now, we will use Middle Term Split of Factorization
➦ 2x² + x - 528 = 0
➦ 2x² - 32x + 33x - 528 = 0
➦ 2x(x-16) + 33(x-16) = 0
➦ (2x+33)(x-16) = 0
Thus,
If x - 16 = 0
➭ x - 16 = 0
➭ x = 16
If 2x + 33 = 0
➭ 2x + 33 = 0
➭ 2x = -33
➭ x = -33/2
We got Length = 16m
Now,
Breadth = (2x+1)
➭ (2 × 16) + 1
➭ 32 +1
➭ 33
Therefore,
Length = 16m
Length = 16mBreadth = 33m
Answer (ii)
Let the two Required Consecutives number be x & x + 1
According to Question
⟹ (x) × (x+1) = 306
⟹ x² + x = 306
⟹ x² + x - 306 = 0
We will do Middle Term Spilit of Factorization
➜ x² + 18x - 17x - 306 = 0
➜ x(x+18) - 17(x+18) = 0
➜ (x-17)(x+18)
Therefore,
If x - 17 = 0
➫ x = 17
And (x+1)
➫ 17 +1
➫ 18
Therefore, required answers are 17 & 18
Answer (iii)
Let Rohan's age be x
Therefore, her mother's age would be (x+26)
After 3 years, age of Rohan = (x+3)
After 3 years, age of Rohan's Mother = (x+26+3) = (x+29)
According to Question
➭ (x+3)(x+29) = 360
➭ x² + 29x + 3x + 87 = 360
➭ x² + 32x + 87 = 360
➭ x² + 32x = 360 - 87
➭ x² + 32x = 273
➦ x² + 32x - 273 = 0
We will do Middle Term Spilit of Factorization
➥ x² + 32x - 273 = 0
➥ x² + 39x - 7x - 273 = 0
➥ x(x+39) -7(x+39) = 0
➥ (x-7)(x+39)
Therefore,
If x - 7 = 0
➫ x - 7 = 0
➫ x = 7
Therefore, Rohan's Present age is 7
Answer (iv)
Let the original speed of Train be x km/hr
Also, distance covered by train is 480 km
As, we know
Time = Distance/Speed
➭ 480/x hr
time = 480/x hrs.
Now, if the train would take 3 hours, then time taken would be 480/x-8
Therefore,
➥ 480/x = (480/x-8 ) + 3
➥ (480/x) - (480/x-8) = 3
After Solving the Equation we got 3x² - 24x + 3480 = 0
➦ 3(3x² - 24x + 3480) = 0
➦ x² - 8x + 1280 = 0
We will do Middle Term Spilit of Factorization
⟹ x² - 8x + 1280 = 0
⟹ x² - 40x + 32x + 1280 = 0
⟹ x(x-40) + 32(x+40) = 0
⟹ (x+32)(x-40)
If x + 32 = 0
➫ x + 32 = 0
➫ x = -32
Speed of train can't be zero, so we we will use next
If x - 40 = 0
➫ x - 40 = 0
➫ x = 40
Therefore, Speed of Train is 40 km/hr
Answer:
1.
let the breadth =b m
let the breadth =b mlength = l =(2b+1)m
let the breadth =b mlength = l =(2b+1)marea =528m²
l*b= 528
(2b+1)b-528=0
2b²+b-528=0
2b²+33b-32b-528=0
b(2b+33)- 16(2b+33)=0
(2b+33)(b-16)=0
2b+33=0 or b-16=0
b should not be zero
therefore
b-16=0
b=16
breadth = 16m
length =l= 2b+1
=2*16+1
=32+1
=33m
2.
The product of two consecutive positive integers is 306.
We need to find the integers
solution :
Let two consecutive numbers are x and (x + 1)
A/C to question,
product of x and (x + 1) = 306
⇒x(x + 1) = 306
⇒x² + x - 306 = 0
⇒ x² + 18x - 17x - 306 = 0
⇒x(x + 18) - 17(x + 18) = 0
⇒(x + 18)(x - 17) = 0⇒ x = 17 and -18
so x = 17 and (x +1) = 18
Therefore the numbers are 17 and 18.
3.
Let Rohan's age be x .......
Therefore Rohan's mom's age is x+26......
Now Roahn age after 3 yrs x+3 and his mothers age after three years x+29......
Now we know (x+3)(x+29) = 360 ...... ...
x²+3x+29x+87 = 360 ......
x²+32x+87=360 .......
x²+32x=360-87 ......
= 273 x(x+32)
=273
After solving we get x=7 So, age of Rohan = 7 yrs
The solution for how to get x is Take one factor of the equation x²+32x-273 = 0
First factorise ...
. Factors are x-7 and x+32 ......
So take any one factor and solve it and you will get two solutions of the quadratic equation.....
So take any one factor and solve it and you will get two solutions of the quadratic equation...... X-7.....x=7 And x+32.....x=-32
So take any one factor and solve it and you will get two solutions of the quadratic equation...... X-7.....x=7 And x+32.....x=-32 Now put your logic age can't be in minus so x=7 is correct solution for this question...Hence solved.......
4.
Distance travelled by the train = 480 km
Let the speed of the train be x kmph
Time taken for the journey = 480/x
Given speed is decreased by 8 kmph
Hence the new speed of train = (x – 8) kmph
Time taken for the journey = 480/(x – 8)
480 - 480 = 3
x-8 x
480 (x-x+8) = 3
x(x-8)
480 × 8= x^2 - 8x
3
1280= x^2- 8x
On solving we get x = 40 Thus the speed of train is 40 kmph..