Which amount is larger and by how much? A positive number a or the same number a decreased by 50% and then increased by 50% of the result?
To determine which amount is larger and by how much, we need to compare the positive number `a` and the same number `a` decreased by 50% and then increased by 50% of the result.
Let's break it down step by step:
1. Start with the positive number `a`.
2. Decrease `a` by 50%: `a - (0.5 * a) = 0.5 * a`.
3. Increase the result by 50% of the decreased value: `0.5 * a + (0.5 * 0.5 * a) = 0.5 * a + 0.25 * a = 0.75 * a`.
Now we have two expressions: `a` and `0.75 * a`. To determine which is larger, we can compare them:
- If `a` is larger than `0.75 * a`, then the amount `a` is larger.
- If `0.75 * a` is larger than `a`, then the amount `a` decreased by 50% and then increased by 50% is larger.
By comparing the two expressions, we can see that `0.75 * a` is smaller than `a` (unless `a` is zero). Therefore, the positive number `a` is larger than the amount obtained by decreasing `a` by 50% and then increasing it by 50% of the result.
To determine how much larger `a` is, we can calculate the difference between the two amounts:
Difference = `a - 0.75 * a = 0.25 * a`.
So, `a` is larger than the other amount by 0.25 * `a`.
a - 50% = .5a
.5a + .5(.5a) = ?