… If some Reds are Blues and all Yellows are Reds, then what must be true? …
Hi Mik. We can think about things other than colors that have the same relationship, if it helps. For example:
All 5G phones are faster phones.
Some faster phones are Android phones.
What does that say about some Android phones? If you're not sure, replace the name 'faster phones' with the name '5G phones'. (We can do that because we've been told that both names mean the same thing.) After you make the substitution, what do you see?
If you'd rather use colors, we can, but don't expect the statements to make sense based on what you know about real color (because red is not the same as yellow). We're told that Reds are the same as Yellows, so let's use the same color to represent both.
\(\quad\) All Reds are Yellows
Some Reds are Blues
In terms of colors you see (not names that you read), what does the last line mean?
It takes 5 window-washers 7 hours to wash one office building. How many hours would it take … 4 window-washers …
We can consider fractional amounts of the whole job. If we do that, then the number 1 represents the whole job.
Five workers each work seven hours to complete the whole job. That's 35 total hours of work that was done. Therefore, each worker completes 1/35th of the job per hour.
Here's another way to see that. If the whole job takes 7 hours to finish, then 1/7th of the job is completed each hour by the crew. That is five workers completing 1/7th of the job (each hour).
1/7 ÷ 5 = 1/35
One worker completes 1/35th of the job each hour.
Therefore, four workers together complete 4/35ths of the job per hour. All of the fractional amounts of the job must add up to 1. In other words:
[fraction of job done by crew per hour] × [number of hours] = 1
4/35 × (?) = 1
How many hours is that ?
If A < B and B + C = 10, then which of the following must be true?
Here's an example with numbers:
If 0 <
1 and 1 + 9 = 10, then it must be true that 0 <
10 - 9
How did I know to replace the expression
1 with the expression
10 - 9?
I looked at the equation 1 + 9 = 10 and realized that I could get a different expression for 1 by removing 9 from each side:
1 = 10 - 9
Look at the equation B + C = 10. If we remove C from each side, we get another expression for B:
B = 10 - C
We don't know what number B is, but now we have two ways to write it. We can write symbol B, or we can write the expression 10-C. Both mean the same number.
We're told that A is smaller than B, and we just found another way to write B. So what must be true also?
A < ?
?
