Answer: 2, 8, 96cm^2, 35 miles
Step-by-step explanation:
What we know:
- We need to solve all of the given problems
- We have the information we need
- We can use ratios to our advantage
How to solve:
By using our knowledge of ratios and substitution, we can solve these problems.
Process:
First problem
Set up ratio 7.5:15
Find GCF (7.5:15)/7.5
Simplify 1:2
Solution: 2
Second problem
Set up equation (x/10) = (6/7.5)
Rewrite (x/10) = (60/75)
Simplify (x/10) = (4/5)
Common denominator (x/10) = (8/10)
Isolate variable [(x/10) = (8/10)]*10
Simplify x = 8
Solution: 8
Third Question
where x represents the new width, y represents the new length, and A represents the new area.
Set up equation A = xy
Substitute A = (2*4)(2*6)
Simplify A = (8)(12)
Simplify A = 96
Solution: 96 cm^2
Fourth Question
Where x represents the unknown length in miles
Set up equation (21:3) = (x:5)
Rewrite in fraction form [(21/3)] /3 = (x/5)
Simplify (7/1) = (x/5)
GCF 5(7/1) = (x/5)
Isolate variable [(35/5) = (x/5)]*5
Solution: 35 miles = x
Answer: 2, 8, 96 cm^2, 35 miles
Check:
Problem 1:
If the scale factor is 2, then the area of a square with the side measurement of 7.5 units should be (due to areas being squared) 1/2^2 of the square with the side measurement of 15 units. Let's solve this:
(1/2^2)(15*15) = (7.5*7.5)
(1/4)(15^2) = (7.5^2)
225/4 = 56.25
56.25 = 56.25
Therefore, a scale factor of 2 is correct.
Problem 2:
If x = 8, then 8/10 should equal 6/7.5. Let's solve this:
when x = 8:
8/10 = 6/7.5
0.8 = 0.8
Therefore, x = 8 is a correct solution.
Problem 3:
If a scale factor of 2 creates the new area of 96 cm^2, then the old area of (6*4) should be (due to areas being squared) 1/2^2 of the new area. Let's solve this:
(1/2^2)(96) = (6*4)
(96/4) = 24
24 = 24
Therefore, the new area is 96 cm^2.
Problem 4:
If the houses are 35 miles apart, then 3 in./21 mi. should be equal to 5 in./35 mi. Let's solve this:
(21/3) = (35/5)
7 = 7
Therefore, the houses are 35 miles apart.