### Ratio word problem solved with block model and algebra

I guess it is time for some more problem solving, since someone sent this question in.

Two numbers are in the ratio of 1:2. If 7 be added to both, their ratio changes to 3:5. What is the greater number?

We can model the two original numbers with blocks. 1 block and 2 blocks makes the ratio to be 1:2.

Now add the same thing to both (the 7):

So the original bigger number, which had two larger blocks, is 28, and the smaller number is 14.

Check:

Their ratio is 28:14 = 2:1. If you add 7 to both, you have 35 and 21, and their ratio is 35:21 = 5:3.

Solving the same problem using algebra

The two numbers in the ratio of 1:2 are x and 2x.

Once 7 is added to both, we have x + 7 and 2x + 7. Their ratio is 3:5, and we can write a proportion using fractions:

Cross-multiply to get

5(x + 7) = 3(2x + 7)

5x + 35 = 6x + 21

35 - 21 = x

x = 14

The larger number was 2x or 28. We already checked this earlier.

Two numbers are in the ratio of 1:2. If 7 be added to both, their ratio changes to 3:5. What is the greater number?

We can model the two original numbers with blocks. 1 block and 2 blocks makes the ratio to be 1:2.

|-------|

|-------|-------|

Now add the same thing to both (the 7):

7The way I just happened to draw these suggests that I could just split the original block in two, and the problem is solved:

|-------|---|

|-------|-------|---|

7

7Here, each little block is 7. The original larger blocks are 14 each.

|---|---|---|

|---|---|---|---|---|

7

So the original bigger number, which had two larger blocks, is 28, and the smaller number is 14.

Check:

Their ratio is 28:14 = 2:1. If you add 7 to both, you have 35 and 21, and their ratio is 35:21 = 5:3.

Solving the same problem using algebra

The two numbers in the ratio of 1:2 are x and 2x.

Once 7 is added to both, we have x + 7 and 2x + 7. Their ratio is 3:5, and we can write a proportion using fractions:

x + 7 3

------- = ----

2x + 7 5

Cross-multiply to get

5(x + 7) = 3(2x + 7)

5x + 35 = 6x + 21

35 - 21 = x

x = 14

The larger number was 2x or 28. We already checked this earlier.