A student claims that a ratio remains unchanged if 1 is added to both the numerator and the denominator of the fraction. Does a/b equal a + 1/b + 1? Write and explanation and give an example or counterexample.
is a and b restricted only to positive numbers (or can they be negative)?
assume that it is true,,, that is,
a/b = (a+1)/(b+1)
ab + a = ab + b
a = b
only true if a = b
e.g.
5/5 = 6/6
To determine whether a ratio remains unchanged if 1 is added to both the numerator and the denominator, let's analyze the claim and provide an explanation.
A ratio is defined as the relationship between two numbers, expressed as a fraction a/b, where 'a' is the numerator and 'b' is the denominator. The student claims that if we add 1 to both the numerator and the denominator, i.e., (a+1)/(b+1), the ratio remains the same.
To verify this claim, let's test it with an example and a counterexample:
Example:
Let's consider the ratio 2/3. According to the student's claim, if we add 1 to both the numerator and the denominator, we would get (2+1)/(3+1) which simplifies to 3/4. To check if the two ratios are equal, we can cross-multiply: 2 * 4 = 3 * 3. However, this equation is false since 8 is not equal to 9. Therefore, in this example, the claim does not hold true.
Counterexample:
To confirm our findings further, let's consider the counterexample where a = 1 and b = 2. According to the student's claim, the ratio 1/2 should be equal to (1+1)/(2+1), which simplifies to 2/3. However, cross-multiplying this equation yields 1 * 3 = 2 * 2, which is false since 3 is not equal to 4. Therefore, in this counterexample as well, the claim does not hold true.
Based on the example and the counterexample, we can conclude that the claim made by the student, a/b = a + 1/b + 1, is false. Adding 1 to both the numerator and the denominator does not result in the same ratio as the original one.