A number obtained by increasing 75% of a given two digit number is 8 more than half of another two digit number formed by reversing the digits of the given number. Then sum of the digits of the given number is
13
10
9
11
I need answer along with the working out part as well
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Step-by-step explanation:
Let's break it down step by step:
Let the two-digit number be represented as 10a + b, where 'a' is the tens digit and 'b' is the ones digit.
The number obtained by increasing 75% of this number: \(1.75 \times (10a + b)\)
The other number formed by reversing the digits: \(10b + a\)
According to the given conditions: \(1.75 \times (10a + b) = 0.5 \times (10b + a) + 8\)
Let's solve it step by step.
Given: \(1.75 \times (10a + b) = 0.5 \times (10b + a) + 8\)
Expanding and solving this equation, we get:
\(17.5a + 1.75b = 5b + 0.5a + 8\)
Grouping 'a' terms on one side and 'b' terms on the other:
\(17.5a - 0.5a = 5b - 1.75b + 8\)
\(17a = 3.25b + 8\)
\(17a - 8 = 3.25b\)
Substituting possible values for 'a' (as it's a ten's digit, it must be an integer) to find 'b', and then find the sum of digits of the number formed.
Let's check the options:
- **Option 1: 13**
- \(17 \times 13 - 8 = 3.25b\)
- \(221 - 8 = 3.25b\)
- \(213 = 3.25b\)
- This doesn't give a whole number for 'b'.
- **Option 2: 10**
- \(17 \times 10 - 8 = 3.25b\)
- \(170 - 8 = 3.25b\)
- \(162 = 3.25b\)
- \(b = 162 / 3.25 = 50\)
- This gives a whole number for 'b' (50).
- Sum of the digits in the number \(10\) is \(1+0 = 1\).
Therefore, the sum of the digits of the given number is \(1\).
Answer:
Let's solve the problem step by step.
Let's assume the given two-digit number is represented as "10a + b," where "a" represents the tens digit and "b" represents the units digit.
According to the problem, a number obtained by increasing 75% of the given two-digit number is 8 more than half of another two-digit number formed by reversing the digits of the given number.
The number obtained by increasing 75% of the given two-digit number is:
1.75 * (10a + b) = 17.5a + 1.75b
The other two-digit number formed by reversing the digits of the given number is:
10b + a
Half of this number is:
0.5 * (10b + a) = 5b + 0.5a
According to the problem, the first number is 8 more than half of the second number:
17.5a + 1.75b = 5b + 0.5a + 8
Now, let's simplify and solve the equation:
17.5a - 0.5a = 5b - 1.75b + 8
17a = 3.25b + 8
17a - 3.25b = 8
To find the sum of the digits of the given number, we need to determine the values of "a" and "b" that satisfy the equation.
Since the given number is a two-digit number, "a" cannot be 0. Let's try different values of "a" and find the corresponding value of "b" that satisfies the equation.
If we assume "a" as 1, we have:
17 - 3.25b = 8
-3.25b = 8 - 17
-3.25b = -9
b = -9 / -3.25
b ≈ 2.769
Since "b" should be a whole number, this solution is not valid.
Let's assume "a" as 2:
34 - 3.25b = 8
-3.25b = 8 - 34
-3.25b = -26
b = -26 / -3.25
b = 8
Now, we have "a" as 2 and "b" as 8. The given number is 28, and the sum of its digits is 2 + 8 = 10.
Therefore, the sum of the digits of the given number is 10.