Kargil's temperature was recorded in a week from Monday to Saturday. All reading were in A.P. The sum of temperature of monday and Saturday was 5℃ more than sum of temperature of Tuesday and Saturday. If temperature of wednesday was -30℃ Celsius then find the temperature on the other five days
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Answers
Answers:
Monday = -20℃
Tuesday = -25℃
Thursday = -35℃
Friday = -40℃
Saturday = -45℃
Prerequisites:
- An AP (arithmetic progression) refers to a sequence of numbers where each successive term differs from the term preceding it by a fixed value, and this fixed value is known as the common difference.
- In an AP, 'a' denotes the first term of the AP, 'd' denotes the common difference, 'n' denotes the total number of terms. The position of the nth term is given by a
where a
Step-by-step explanation:
According to the question, the temperatures of 6 days (Mon, Tue, Wed, Thu, Fri, Sat), and their temperatures are in an AP, i.e, arithmetic progression.
An AP of 6 terms goes like this,
- a - 5d, a - 3d, a - d, a + d, a + 3d, a + 5d
Where a - 5d, a - 3d, a - d, a + d, a + 3d and a + 5d stand for the temperature on Monday, Tuesday, Wednesday, Thursday, Friday and Saturday respectively.
Re-written here for readability:
- a - 5d = Temp(Monday)
- a - 3d = Temp(Tuesday)
- a - d = Temp(Wednesday)
- a + d = Temp(Thursday)
- a + 3d = Temp(Friday)
- a + 5d = Temp(Saturday)
The temperature on Wednesday is -30℃, on substituting it we get,
- a - d = -30℃
ATQ, the sum of the temperature on Monday and Saturday was 5℃ more than the sum of the temperature on Tuesday and Saturday.
➝ Temp(Mon) + Temp(Sat) = 5℃ + Temp(Tue) + Temp(Sat)
➝ Temp(Sat) - Temp(Sat) = 5℃ + Temp(Tue) - Temp(Mon)
➝ 0 = 5℃ + Temp(Tue) - Temp(Mon)
➝ 0 = 5℃ + a - 3d - [a - 5d]
➝ 0 = 5℃ + a - 3d - a + 5d
➝ 0 = 5℃ - 3d + 5d
➝ 0 = 5℃ + 2d
➝ -2d = 5℃
➝ d = 5℃/-2
➝ d = -2.5℃
Also ATQ, the temperature on Wednesday was -30℃, therefore,
➝ Temp(Wednesday) = -30℃
➝ a - d = -30℃
On substituting d = -2.5℃ we get,
➝ a - (-2.5) = -30℃
➝ a + 2.5℃ = -30℃
➝ a = -30℃ - 2.5℃
➝ a = -32.5
Now, we have the value of 'a' (the first term), and 'd' (the common difference), this will allow us to find out the temperature on each day.
Monday
➝ a - 5d
➝ -32.5 - 5(-2.5)
➝ -32.5 + 12.5
➝ -20℃
Tuesday
➝ a - 3d
➝ -32.5 - 3(-2.5)
➝ -32.5 + 7.5
➝ -25℃
Thursday
➝ a + d
➝ -32.5 + (-2.5)
➝ -32.5 - 2.5
➝ -35℃
Friday
➝ a + 3d
➝ -32.5 + 3(-2.5)
➝ -32.5 - 7.5
➝ -40℃
Saturday
➝ a + 5d
➝ -32.5 + 5(-2.5)
➝ -32.5 - 12.5
➝ -45℃
Therefore the temperature recorded on Monday, Tuesday, Thursday, Friday, Saturday are -20℃, -25℃, -35℃, -40℃ and -45℃ respectively.
Verification:
We know that wednesday's temperature is -30℃, so if we substitute 'a' and 'd' in 'a - d', we should get 30℃ if the values we've obtained as 'a' and 'd' are correct.
➝ a - d = -30℃
➝ -32.5 - (-2.5) = -30℃
➝ -32.5 + 2.5 = -30℃
➝ -30℃ = -30℃
➝ LHS = RHS.
Hence the values we've obtained for 'a' and 'd' are correct.
We can also verify our final answers by substituting them in the relation that says the sum of the temperature on Monday and Saturday was 5℃ more than the sum of the temperature on Tuesday and Saturday.
➝ Temp(Mon) + Temp(Sat) = 5℃ + Temp(Tue) + Temp(Sat)
➝ -20℃ + (-45℃) = 5℃ + (-25℃) + (-45℃)
➝ -20℃ - 45℃ = 5℃ - 25℃ - 45℃
➝ -65℃ = -20℃ - 45℃
➝ -65℃ = -65℃
➝ LHS = RHS.
Hence our final values are correct as well.
Answer:
Since the temperature at hill station from Monday to Friday is in A.P.
∴ Let us assume those 5 days temperature be
Monday = a – 2d
Tuesday = a – d
Wednesday = a
Thursday = a + d
Friday = a + 2d
According to first condition
a – 2d + a – d + a = 0
∴ 3a – 3d = 0 ------------- eq. no. (1)
According to second condition
a + d + a + 2d = 15
∴ 2a + 3d = 15 -------------eq. no. (2)
Adding eq. (1) and eq. (2)
3a – 3d = 0
2a + 3d = 15
5a + 0 = 15
∴ a = 15/5
∴ a = 3
Substituting a = 3 in eq. (2)
2a + 3d = 15
∴ 2(3) + 3d = 15
∴ 6 + 3d = 15
∴ 3d = 15 – 6
∴ 3d = 9
∴ d = 9/3
∴ d = 3
Monday = a – 2d = 3 – 2(3) = 3 – 6 = - 30 C
Tuesday = a – d = 3 – 3 = 00 C
Wednesday = a = 3 0 C
Thursday = a + d = 3 + 3 = 60 C
Friday = a + 2d = 3 + 2(3) = 3 + 6 = 90 C
∴ The temperature from Monday to Friday are
- 3 0 C, 00 C, 30 C, 60C and 90C respectively.
Step-by-step explanation:
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