35. If b is postive integers less than 400 and more than 100, then how many integer pairs (a,b) satisify the equation a/b=2/9?
To find the number of integer pairs (a,b) that satisfy the equation a/b = 2/9, we can use the following steps.
Step 1: Determine the range of values for b.
Given that b is a positive integer less than 400 and greater than 100, we know that b ranges from 101 to 399 (inclusive).
Step 2: Determine the possible values for a.
To satisfy the equation a/b = 2/9, we need to find positive integer values for a that, when divided by b, result in the fraction 2/9.
Since the numerator and denominator of the fraction are 2 and 9 respectively, a multiple of 2 must divide evenly into b, and a multiple of 9 must divide evenly into a.
Step 3: Find the common multiples of 2 and 9.
To find the common multiples of 2 and 9, we can list out the multiples of each number separately and find their common multiples.
Multiples of 2: 2, 4, 6, 8, ...
Multiples of 9: 9, 18, 27, 36, ...
Common multiples: 18, 36, 54, 72, ...
Step 4: Determine the valid pairs of (a,b).
Since b divides evenly by 2, and a divides evenly by 9, we can use the common multiples we found to match values of b with valid values of a to satisfy the equation.
For each common multiple, we check if it falls within the range of b values (101 to 399). If it does, we consider it a valid pair and increment the count.
Step 5: Calculate the count of valid pairs.
By going through the range of b values (101 to 399) and matching them with the valid values of a (common multiples of 2 and 9), we can count the number of valid pairs that satisfy the equation a/b = 2/9.
To summarize, follow these steps:
1. Determine the range of values for b (101 to 399).
2. Find the common multiples of 2 and 9.
3. Match the valid values of a with the range of b values.
4. Count the number of valid pairs.
Note: It may be helpful to use a programming language or a spreadsheet to automate the process and obtain the final count of valid pairs.