Linear Equation
- The difference between the squares of two numbers is 256000 and the sum of the numbers is 1000. The numbers are
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Let us assume the numbers are a and b.
According to question,
a2 - b2 = 256000 .......................(1)
and a + b = 1000 .........................(2)
Solve the above equation.Correct Option: B
Let us assume the numbers are a and b.
According to question,
a2 - b2 = 256000 .......................(1)
and a + b = 1000 .........................(2)
On dividing the Equation (1) with Equation (2),
(a2 - b2)/(a + b) = 256000/1000
(a2 - b2)/(a + b) = 256
(a + b)(a - b)/(a + b) = 256
a - b = 256...............................(3)
Add the equation (2) and (3), we will get
a + b + a - b = 1000 + 256
2a = 1256
a = 628
Put the value of a in equation (2), we will get
658 + b = 1000
b = 1000 - 628
b = 372
So answer is 628 , 372.
- Two-fifths of one -fourth of three -seventh of a number is 15. What is half of that number?
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Let us assume the number is P.
According the question,
three -seventh of a number = 3/7 x P
one -fourth of three -seventh of a number = 1/4 x 3/7 x P
Two-fifths of one -fourth of three -seventh of a number = 2/5 x 1/4 x 3/7 x P
According to question,
Two-fifths of one -fourth of three -seventh of a number = 15
2/5 x 1/4 x 3/7 x P = 15
Solve the above equation.Correct Option: D
Let us assume the number is P.
According the question,
three -seventh of a number = 3/7 x P
one -fourth of three -seventh of a number = 1/4 x 3/7 x P
Two-fifths of one -fourth of three -seventh of a number = 2/5 x 1/4 x 3/7 x P
⇒ 2/5 x 1/4 x 3/7 x P = 15
⇒ 2 x 1 x 3 x P = 15 x 7 x 4 x 5
⇒ P = 15 x 7 x 4 x 5 / 2 x 3
⇒ P = 5 x 7 x 2 x 5
⇒ P = 350
∴ P /2 = 350/2
⇒ P /2 = 175
- The numbers are such that the square of one is 224 less than 8 times the square of the other. If the numbers be in the ratio 3 : 4, the numbers are
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Let us assume the ratio term is r.
then the number will be 3r and 4r.
According to question,
(4r)2 + 224 = 8 x (3r)2
Solve the equation and find the answer.Correct Option: C
Let us assume the ratio term is r.
then the number will be 3r and 4r.
According to question,
(4r)2 + 224 = 8 x (3r)2
16r2 + 224 = 8 x 9r2
16r2 + 224 = 72r2
72r2 - 16r2 = 224
56r2 = 224
r2 = 224/56
r2 = 4
r = 2
First number = 3r = 3 x 2 = 6
Second number = 4r = 4 x 2 = 8
- A person on tour has total ₹360 for his daily expenses. He decides to extend his tour programme by 4 days which leads to cutting down daily expenses by ₹3 a day . The numbers of days of tour are :
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Method 1 to solve this question.
Let us assume the numbers of days of tour before extending the tour are D and daily expenses is E.
According to question,
Total expenses on tour = per day expenses x total number of days
Total expenses on tour = E x D = ED
360 = ED
Again according to question,
After extending the tour by 4 days, Number of days of tour = D + 4
Expenses will be reduced by 3 rupees, then everyday expenses = E - 3
so total expenses = (D + 4) x (E - 3)
Solve the equation and get the answer.
Method 2 to solve this question.
Let us assume the original day of tour is d days.
Given that his tour is extended for 4 days
Hence daily expenses per days = 360/(d + 4)
Therefore, According to question,
360/d - 360/(d + 4) = 3
Solve the equation and get the answer.Correct Option: B
Method 1 to solve this question.
Let us assume the numbers of days of tour before extending the tour are D and daily expenses is E.
According to question,
Total expenses on tour = per day expenses x total number of days
Total expenses on tour = E x D = ED
360 = ED
⇒ ED = 360
⇒ E = 360/D .................(1)
Again according to question,
After extending the tour by 4 days, Number of days of tour = D + 4
Expenses will be reduced by 3 rupees, then everyday expenses = E - 3
so total expenses = (D + 4) x (E - 3)
360 = (D + 4) x (E - 3)
(D + 4) x (E - 3) = 360 .............................................(2)
Put the value of E in Equation (2). we will get,
(D + 4) x (360/D - 3) = 360
⇒ (D + 4) x ( (360 - 3D)/D ) = 360
⇒ (D + 4) x ( (360 - 3D) ) = 360 x D
⇒ (D + 4) x (360 - 3D) = 360 x D
After multiplication by algebra law,
⇒ 360 x D - 3D x D + 4 x 360 - 4 x 3D = 360D
⇒ 360 x D - 3D+ 1440 - 12D = 360D
⇒ - 3D+ 1440 - 12D = 360D - 360 x D
⇒ - 3D+ 1440 - 12D = 0
⇒ 3D- 1440 + 12D = 0
⇒ D- 480 + 4D = 0
⇒ D+ 4D - 480 = 0
⇒ D+ 24D - 20D - 480 = 0
⇒ D(D + 24) - 20(D + 24) = 0
⇒ (D + 24) (D - 20) = 0
Either (D + 24) = 0 or (D - 20) = 0
So D = - 24 or D = 20
But days cannot be negative so D = 20 days.
Method 2 to solve this question.
Let us assume the original day of tour is d days.
Given that his tour is extended for 4 days
Hence daily expenses per days = 360/(d + 4)
Therefore, According to question,
360/d - 360/(d + 4) = 3
⇒ (360(d + 4) - 360d)/d x (d + 4) = 3
⇒ (360(d + 4) - 360d)/d x (d + 4) = 3
⇒ 360(d + 4) - 360d = 3d x (d + 4)
⇒ 360d + 1440 – 360d = 3(d2 + 4d)
⇒ 1440 = 3d2 + 12d
⇒ 3d2 + 12d – 1440 = 0
⇒ d2 + 4d – 480 = 0
⇒ d2 + 24d – 20d – 480 = 0
⇒ d(d + 24) – 20(d + 24) = 0
⇒ (d + 24)(d – 20) = 0
⇒ (d + 24) = 0 or (d – 20) = 0
? d = –24 or d = 20
Since days cannot be negative, d = 20
Hence his original duration of the tour is 20 days.
- The sum of three consecutive odd numbers is 20 more than the first of these numbers.What is the middle number ?
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Let us assume the first Odd number is a.
then 2nd odd number = a + 2
and 3rd odd number = a + 4
According to question,
Sum of three consecutive odd numbers is 20 more than the first of these numbers;
a + (a + 2) + (a + 4) = a + 20
Solve the equation and get the answer.Correct Option: B
Let us assume the first Odd number is a.
then 2nd odd number = a + 2
and 3rd odd number = a + 4
According to question,
Sum of three consecutive odd numbers is 20 more than the first of these numbers;
a + (a + 2) + (a + 4) = a + 20
⇒ a + a + 2 + a + 4 - a = 20
⇒ 2a + 6 = 20
⇒ 2a + 6 = 20
⇒ 2a = 20 - 6
⇒ 2a = 14
⇒ a = 7
The First odd number a = 7
The second odd number = a + 2 = 7 + 2 = 9
Second odd number is middle number = 9