Find two positive decimals a and b that make the following statement true. Then find two positive decimals a and b that make the statement false.
If a < 1 and b < 1, then a ÷ b < 1.
case1
let a = .25, and b = .3
a+b = .55 which is < 1
case 2:
let a = .8, and b = .7
then a+b = 1.5 which is NOT < 1
the statement should have been
if a<1 and b< 1, then a+b < 2
(by adding the 2 inequalities)
To find two positive decimals a and b that make the statement true, we need to ensure that both a and b are less than 1, and their division, a ÷ b, is also less than 1.
Let's try a = 0.5 and b = 0.25.
Here, a and b are both less than 1 (0.5 < 1 and 0.25 < 1). When we divide a by b, we get:
a ÷ b = 0.5 ÷ 0.25 = 2
Since 2 is greater than 1, this example does not satisfy the condition. We need to choose smaller values for a and b.
Let's try a = 0.2 and b = 0.1.
Here, a and b are both less than 1 (0.2 < 1 and 0.1 < 1). When we divide a by b, we get:
a ÷ b = 0.2 ÷ 0.1 = 2
Since 2 is greater than 1, this example also does not satisfy the condition. We need to choose even smaller values for a and b.
Let's try a = 0.1 and b = 0.05.
Here, a and b are both less than 1 (0.1 < 1 and 0.05 < 1). When we divide a by b, we get:
a ÷ b = 0.1 ÷ 0.05 = 2
Again, 2 is greater than 1, so this example does not satisfy the condition. Let's try even smaller values for a and b.
Finally, let's try a = 0.01 and b = 0.005.
Here, a and b are both less than 1 (0.01 < 1 and 0.005 < 1). When we divide a by b, we get:
a ÷ b = 0.01 ÷ 0.005 = 2
Surprisingly, 2 is again greater than 1, and we are not able to find two positive decimals a and b that make the statement true.
Now, let's find two positive decimals a and b that make the statement false. In this case, we need to find a pair of decimals a and b, where a and b are both less than 1, but their division, a ÷ b, is greater than or equal to 1.
For example, let's try a = 0.5 and b = 0.2.
Here, a and b are both less than 1 (0.5 < 1 and 0.2 < 1). When we divide a by b, we get:
a ÷ b = 0.5 ÷ 0.2 = 2.5
Since 2.5 is greater than 1, this example satisfies the condition, and a and b are positive decimals that make the statement false.
To summarize:
True Example: No positive decimals a and b satisfy the condition.
False Example: a = 0.5 and b = 0.2.