This one, I am very confused on and I have no idea how to solve. Please help explain it so I can understand it.
Given that the positive integers a, b, c, and d satisfy a/b < c/d < 1, arrange the following in order of increasing magnitude: b/a, d/c, bd/ac, b + d/ a + c, 1.
Thanks in advance!
Think it out.
Suppose c were 5 and d 9
and a were 1 and b were 9
So 1/9<5/9<1, right?
so b/a becomes 9, d/c becomes 1.8, bd/ac becomes 81/5 or 16.x, and you can do the others, ranking them. Will it hold for other numbers?
How did you pick those numbers? Randomly, or was there a method?
He picked the numbers according to your rules.
To solve this problem, we need to understand the given inequality and compare the magnitudes of the expressions with respect to increasing values. Let's break it down step-by-step:
1. First, let's understand the given inequality: a/b < c/d < 1
This means that the fraction a/b is less than the fraction c/d, which is less than 1. This sets a constraint on the values of a, b, c, and d.
2. Now, let's compare the magnitude of b/a:
To compare the magnitude of b/a, we need to understand that as the numerator (b) gets larger and the denominator (a) gets smaller, the magnitude of the fraction b/a increases. So, b/a will be smaller in magnitude than 1 (since the inequality condition is a/b < 1).
3. Next, let's compare the magnitude of d/c:
Similar to b/a, we need to understand that as the numerator (d) gets larger and the denominator (c) gets smaller, the magnitude of the fraction d/c increases. So, d/c can be smaller or greater than 1, depending on the values of d and c.
4. Moving on to bd/ac:
To compare the magnitude of bd/ac, we can think in terms of cross-multiplication. If the numerator (bd) gets larger and the denominator (ac) gets smaller, the magnitude of the fraction bd/ac increases. So, bd/ac can be smaller or greater than 1, depending on the values of b, d, a, and c.
5. Now, let's compare the magnitude of (b + d)/(a + c):
To compare the magnitude of (b + d)/(a + c), we need to consider the sum of the numerators (b + d) and the sum of the denominators (a + c). If the numerator gets larger and the denominator gets smaller, the magnitude of the fraction (b + d)/(a + c) increases. So, (b + d)/(a + c) can be smaller or greater than 1, depending on the values of b, d, a, and c.
6. Finally, let's compare the magnitude of 1:
Since 1 is a whole number, it will always have a larger magnitude than any fraction that is less than 1. Therefore, 1 will have the largest magnitude.
To summarize, in order of increasing magnitude:
b/a < d/c < bd/ac < (b + d)/(a + c) < 1