a. If 2 is added to the numerator and denominator of a fraction, it becomes [tex]$\frac{9}{10}$[/tex] and if 3 is subtracted from the same numerator and denominator, it becomes [tex]$\frac{4}{5}$[/tex]. Find the fraction.
b. Write [tex]$1011.101_2$[/tex] as base 10 number.
Answer (2)
The fraction derived from the given conditions is 8 7 and the base 10 equivalent of the binary number 1011.10 1 2 is 11.625 .
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Set up a system of equations based on the given conditions for the fraction.
Solve the system of equations to find the numerator and denominator of the fraction.
Convert the binary number to base 10 by expanding in powers of 2.
The fraction is 8 7 and the base 10 equivalent is 11.625 .
Explanation
Problem Analysis Let's break down this problem step by step. First, we need to find a fraction based on the given conditions. Then, we'll convert a binary number to its decimal equivalent.
Setting up Equations Let the fraction be y x . According to the problem, adding 2 to both the numerator and denominator gives us 10 9 , and subtracting 3 from both gives us 5 4 . This gives us two equations: y + 2 x + 2 = 10 9 y − 3 x − 3 = 5 4
Solving the First Equation Let's solve the first equation for x and y . Cross-multiplying gives us: 10 ( x + 2 ) = 9 ( y + 2 ) 10 x + 20 = 9 y + 18 10 x − 9 y = − 2
Solving the Second Equation Now, let's solve the second equation. Cross-multiplying gives us: 5 ( x − 3 ) = 4 ( y − 3 ) 5 x − 15 = 4 y − 12 5 x − 4 y = 3
Solving the System of Equations We now have a system of two equations: 10 x − 9 y = − 2 5 x − 4 y = 3
To solve this system, we can multiply the second equation by 2: 2 ( 5 x − 4 y ) = 2 ( 3 ) 10 x − 8 y = 6
Finding y Now, subtract the first equation from the modified second equation: ( 10 x − 8 y ) − ( 10 x − 9 y ) = 6 − ( − 2 ) y = 8
Finding x Substitute y = 8 into the equation 5 x − 4 y = 3 :
5 x − 4 ( 8 ) = 3 5 x − 32 = 3 5 x = 35 x = 7
The Fraction So, the fraction is 8 7 .
Converting Binary to Decimal Now, let's convert the binary number 1011.10 1 2 to base 10. We can express it 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 ⋅ 0.5 + 0 ⋅ 0.25 + 1 ⋅ 0.125 = 8 + 0 + 2 + 1 + 0.5 + 0 + 0.125 = 11.625
The Decimal Equivalent Therefore, 1011.10 1 2 = 11.62 5 10 .
Final Answer In summary, the fraction is 8 7 and the base 10 equivalent of 1011.10 1 2 is 11.625 .
Examples
Understanding fractions and binary numbers is crucial in many real-world applications. For example, when you're cooking and need to halve a recipe, you're using fractions. In computer science, binary numbers are the foundation of how computers store and process information. Converting between binary and decimal helps in understanding how data is represented in digital systems, from the simplest calculations to complex algorithms.
