If \(\frac b a = 0.7,\) find the value of \(\frac {a-b}{a+b} + \frac {11}{34}.\)
0.5
Solving Algebraic Expressions with Given Ratios
We are asked to find the value of the expression \( \frac {a-b}{a+b} + \frac {11}{34} \) given that \( \frac b a = 0.7 \). This problem involves evaluating an algebraic expression using a given ratio.
Understanding the Given Ratio
The problem provides the ratio of \(b\) to \(a\):
\( \frac b a = 0.7 \)
This ratio tells us the relationship between the variables \(a\) and \(b\). We can write this as \( b = 0.7a \).
Evaluating the First Part of the Expression: \( \frac {a-b}{a+b} \)
We need to evaluate \( \frac {a-b}{a+b} \). We can use the relationship \( b = 0.7a \) by substituting \(0.7a\) for \(b\) in this fraction:
\( \frac {a-b}{a+b} = \frac {a - (0.7a)}{a + (0.7a)} \)
Now, simplify the numerator and the denominator:
- Numerator: \( a - 0.7a = (1 - 0.7)a = 0.3a \)
- Denominator: \( a + 0.7a = (1 + 0.7)a = 1.7a \)
So, the expression becomes:
\( \frac {0.3a}{1.7a} \)
Assuming \(a \neq 0\) (if \(a=0\), then \(b=0\), and \(\frac{b}{a}\) would be undefined), we can cancel out the \(a\) from the numerator and the denominator:
\( \frac {0.3}{1.7} \)
To remove the decimals, multiply both the numerator and the denominator by 10:
\( \frac {0.3 \times 10}{1.7 \times 10} = \frac {3}{17} \)
Alternatively, we could divide the numerator and denominator of \( \frac {a-b}{a+b} \) by \(a\):
\( \frac {\frac{a-b}{a}}{\frac{a+b}{a}} = \frac {\frac{a}{a} - \frac{b}{a}}{\frac{a}{a} + \frac{b}{a}} \)
Substitute the given ratio \( \frac{b}{a} = 0.7 \):
\( \frac {1 - \frac{b}{a}}{1 + \frac{b}{a}} = \frac {1 - 0.7}{1 + 0.7} = \frac {0.3}{1.7} = \frac{3}{17} \)
Both methods give the same result for the first part of the expression.
Adding the Second Part of the Expression
Now we need to add \( \frac {11}{34} \) to the result from the first part, which is \( \frac {3}{17} \):
Total Expression \( = \frac {3}{17} + \frac {11}{34} \)
To add these fractions, we need a common denominator. The least common multiple of 17 and 34 is 34.
Convert \( \frac{3}{17} \) to an equivalent fraction with a denominator of 34:
\( \frac {3}{17} = \frac {3 \times 2}{17 \times 2} = \frac {6}{34} \)
Now add the fractions:
\( \frac {6}{34} + \frac {11}{34} = \frac {6 + 11}{34} = \frac {17}{34} \)
Simplifying the Final Fraction
The resulting fraction is \( \frac {17}{34} \). Both the numerator and the denominator are divisible by 17.
\( \frac {17}{34} = \frac {17 \div 17}{34 \div 17} = \frac {1}{2} \)
Converting to Decimal
The question's options are in decimal form. Convert the final fraction \( \frac{1}{2} \) to a decimal:
\( \frac {1}{2} = 0.5 \)
The value of the expression \( \frac {a-b}{a+b} + \frac {11}{34} \) is 0.5.
Summary of Calculation Steps
| Step | Calculation | Result |
|---|---|---|
| Start with given ratio | \( \frac{b}{a} = 0.7 \) | \( b = 0.7a \) |
| Evaluate first part \( \frac{a-b}{a+b} \) | Substitute \( b = 0.7a \) | \( \frac{a-0.7a}{a+0.7a} = \frac{0.3a}{1.7a} \) |
| Simplify the fraction | Cancel \(a\) and remove decimals | \( \frac{0.3}{1.7} = \frac{3}{17} \) |
| Add the second part \( \frac{11}{34} \) | \( \frac{3}{17} + \frac{11}{34} \) | Find common denominator (34): \( \frac{6}{34} + \frac{11}{34} \) |
| Add fractions | \( \frac{6+11}{34} \) | \( \frac{17}{34} \) |
| Simplify fraction | Divide numerator/denominator by 17 | \( \frac{1}{2} \) |
| Convert to decimal | \( \frac{1}{2} \) as decimal | \( 0.5 \) |
Revision Table: Key Concepts for Algebraic Expressions
| Concept | Explanation | Application in this problem |
|---|---|---|
| Ratio | A comparison of two quantities by division. \( \frac{b}{a} = 0.7 \) means \(b\) is 0.7 times \(a\). | Used to establish the relationship \(b = 0.7a\). |
| Algebraic Substitution | Replacing a variable with an expression it is equal to. | Substituting \(0.7a\) for \(b\) in the expression \( \frac{a-b}{a+b} \). |
| Simplifying Fractions | Reducing a fraction to its lowest terms by dividing the numerator and denominator by their greatest common divisor. | Simplifying \( \frac{17}{34} \) to \( \frac{1}{2} \). |
| Adding Fractions | To add fractions, they must have a common denominator. Convert fractions to equivalent fractions with the common denominator, then add numerators. | Adding \( \frac{3}{17} \) and \( \frac{11}{34} \) by using 34 as the common denominator. |
| Decimal Conversion | Converting a fraction into its decimal representation by dividing the numerator by the denominator. | Converting \( \frac{1}{2} \) to 0.5. |
Additional Information: Ratios, Fractions, and Decimals
Ratios, fractions, and decimals are different ways to represent parts of a whole or relationships between quantities. Understanding how to convert between them is essential in many math problems.
- Ratios: Can be written as \(a:b\), \(a/b\), or "a to b". The ratio \( \frac{b}{a} = 0.7 \) is equivalent to \( \frac{7}{10} \). This means for every 10 units of \(a\), there are 7 units of \(b\). For example, if \(a=10\), then \(b=7\); if \(a=20\), then \(b=14\).
- Fractions: Represent parts of a whole or ratios. They consist of a numerator and a denominator. In this problem, we dealt with fractions like \( \frac{3}{17} \) and \( \frac{11}{34} \). Adding fractions requires finding a common denominator.
- Decimals: Another way to represent fractions, especially those with denominators that are powers of 10. The decimal 0.7 is equal to the fraction \( \frac{7}{10} \). The decimal 0.5 is equal to \( \frac{5}{10} \) or \( \frac{1}{2} \).
Being comfortable with these conversions and operations (like adding fractions or substituting values in expressions) is key to solving such quantitative aptitude problems.
Important Questions from Componendo or Dividendo
Consider the following statements:
1. If (a + b) is directly proportional to (a - b), then (a2 + b2) is is directly proportional to ab.
2. If a is directly proportional to b, then (a2 - b2) is directly proportional to ab.
Which of the statements given above is/are correct?
What is \(\rm \frac{x^2+a b}{x^2+m^2 a b}\) equal to?
If \(\rm \frac{a+b}{b+c}=\frac{c+d}{d+a}\) a ≠ c, then which one of the following is correct ?
If \(\rm\frac{\sqrt{x+20}+\sqrt{x-1}}{\sqrt{x+20}-\sqrt{x-1}}=\frac{7}{3}\) , then what is the value of \(\rm \sqrt{(x + 20)(x-1)}\) ?
For \(x = \frac{{4\sqrt 6 }}{{\sqrt 2 + \sqrt 3 }},\) what is the value of \(\frac{{x + 2\sqrt 2 }}{{x - 2\sqrt 2 }} + \frac{{x\; + \;2\sqrt 3 }}{{x - 2\sqrt 3 }}?\)
