Select all the ratios that are equivalent to the ratio 12 : 3. explain how you know.
a. 6: 1
b. 1: 4
c. 4: 1
d. 24 : 6
e. 15: 6
f. 1,200 : 300
g. 112: 13

the answers are c, d, &f

Step-by-step explanation:

the equivalence to 12:3

1200:300. = 12:3 (reduce by 100)

12:3= 4:1 (reduce by 3)

24:6= 12:3 (reduce by 2)

The ratio 12: 3 between two quantities A and B means that 12 times A is 3 times B, or 12*A = 3*B.

Then, if we multiply both sides by the same number k, we got:

k*12*A = k*3*B

If K = 2, then 24*A = 6*B, so the option D; 24:6 is equivalent.

if K = 1/3, then: 4*A = B, so the option C; 4:1 is equivalent.

if k = 100, then : 1200*A = 300*B, so the option F; 1200:300 is equivalent.

You can notice that in all the cases the ratio 12:3 is multiplied in both sides by the same number k, this is the relation you need to search for.

There will be one solution because the lines intersect at exactly one set of points.

(0,6)(6,3)
slope(m) = (3-6) / (6-0) = -3/6 = -1/2

y = mx + b
6 = -1/2(0) + b
6 = b
y = -1/2x + 6

(0,0)(6,6)
slope(m) = (6-0) / (6-0) = 6/6 = 1

y = mx + b
6 = 1(6) + b
6 = 6 + b
6 - 6 = b
0 = b

y = x + 0

x = -1/2x + 6
1/2x + x = 6
1/2x + 2/2x = 6
3/2x = 6
x = 6/(3/2)
x = 6 * 2/3
x = 12/3 = 4

solution is : (4,4)