A man gets a monthly pay of $Nx$. His monthly rent is $8,000.00. After paying his rent, he is left with less than 20,000. Find the range of values of $x$.
If 2 is added to the numerator and denominator of a fraction, it becomes $\frac{9}{10}$ and if 3 is subtracted from the same numerator and denominator, it becomes $\frac{4}{5}$. Find the fraction. Write $1011.101_2$ as base 10 number.
Answer (2)
The range of values for x is x < N 28000 , the fraction found is 8 7 , and the base 10 equivalent of 1011.10 1 2 is 11.625 .
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Determine the range of x by setting up the inequality N x − 8000 < 20000 and solving for x , resulting in x < N 28000 .
Find the fraction by setting up a system of equations b + 2 a + 2 = 10 9 and b − 3 a − 3 = 5 4 , then solving for a and b to get the fraction 8 7 .
Convert the base 2 number 1011.10 1 2 to base 10 by expanding it as 1 ⋅ 2 3 + 0 ⋅ 2 2 + 1 ⋅ 2 1 + 1 ⋅ 2 0 + 1 ⋅ 2 − 1 + 0 ⋅ 2 − 2 + 1 ⋅ 2 − 3 , which equals 11.625 .
The final answers are: x < N 28000 , the fraction is 8 7 , and the base 10 number is 11.625 .
Explanation
Problem Analysis We are given three problems: finding the range of values for x based on a man's monthly pay and rent, finding a fraction given information about adding and subtracting from its numerator and denominator, and converting a base 2 number to base 10.
Finding the Range of x First, let's find the range of values for x . The man's monthly pay is N x , and his rent is $8000. After paying rent, he has less than $20000 left. This can be written as the inequality:
N x − 8000 < 20000
Adding 8000 to both sides, we get:
N x < 28000
Dividing both sides by N , we find the range of values for x :
x < N 28000
Finding the Fraction Next, let's find the fraction. Let the fraction be b a . We are given two conditions:
If 2 is added to the numerator and denominator, the fraction becomes 10 9 .
If 3 is subtracted from the numerator and denominator, the fraction becomes 5 4 .
This gives us the following system of equations:
b + 2 a + 2 = 10 9
b − 3 a − 3 = 5 4
Cross-multiplying the first equation, we get:
10 ( a + 2 ) = 9 ( b + 2 )
10 a + 20 = 9 b + 18
10 a − 9 b = − 2 (Equation 1)
Cross-multiplying the second equation, we get:
5 ( a − 3 ) = 4 ( b − 3 )
5 a − 15 = 4 b − 12
5 a − 4 b = 3 (Equation 2)
Now we can solve this system of equations. Multiply Equation 2 by 2:
10 a − 8 b = 6 (Equation 3)
Subtract Equation 1 from Equation 3:
( 10 a − 8 b ) − ( 10 a − 9 b ) = 6 − ( − 2 )
b = 8
Substitute b = 8 into Equation 2:
5 a − 4 ( 8 ) = 3
5 a − 32 = 3
5 a = 35
a = 7
So the fraction is 8 7 .
Converting Base 2 to Base 10 Finally, let's convert the base 2 number 1011.10 1 2 to base 10. We can write this as:
1 ⋅ 2 3 + 0 ⋅ 2 2 + 1 ⋅ 2 1 + 1 ⋅ 2 0 + 1 ⋅ 2 − 1 + 0 ⋅ 2 − 2 + 1 ⋅ 2 − 3
= 1 ⋅ 8 + 0 ⋅ 4 + 1 ⋅ 2 + 1 ⋅ 1 + 1 ⋅ 2 1 + 0 ⋅ 4 1 + 1 ⋅ 8 1
= 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125
= 11 + 0.5 + 0.125
= 11.625
Final Answer Therefore, the range of values for x is x < N 28000 , the fraction is 8 7 , and the base 10 representation of 1011.10 1 2 is 11.625 .
Examples
Understanding ranges and inequalities is crucial in personal finance. For instance, if you earn a certain amount and have fixed expenses like rent, knowing the range of your discretionary spending helps you budget effectively. Similarly, converting between number systems is essential in computer science, where binary (base-2) is fundamental for representing data. Fractions are used in everyday life, from cooking to measuring ingredients, making it important to understand how to manipulate them.
