# Jerry is 50 inches tall growing at a rate of 1/24 inches per month. Adam is 47 inches tall growing at a rate of 1/8 per month. If they continue to grow at these rates for the next 4 years in how many months will they be the same height? Solve two different ways and show solution processes

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## Jerry-height = 50 + (1/24)m

they have the same height when ...

50 + m/24 = 47 + m/8
3 = m/8 - m/24
m/12 = 3
m = 36

or, from

50 + m/24 = 47 + m/8
multiply each term by 24 to get rid of fractions
1200 + m = 1128 = 3m
72 = 2m
m = 36

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## To find out when Jerry and Adam will be the same height, we can set up an equation based on their growth rates.

First, let's set up the equation for Jerry's height after t months:
J(t) = 50 + (1/24)t

Next, let's set up the equation for Adam's height after t months:
A(t) = 47 + (1/8)t

Now, we need to solve for t when J(t) = A(t):
50 + (1/24)t = 47 + (1/8)t

Let's solve this equation algebraically:

1. Multiply both sides of the equation by 24 to eliminate the fraction:
24(50) + t = 24(47) + 3t

2. Simplify the equation:
1200 + t = 1128 + 3t

3. Subtract t from both sides of the equation:
1200 = 1128 + 2t

4. Subtract 1128 from both sides of the equation:
72 = 2t

5. Divide both sides of the equation by 2:
t = 36

So, after 36 months, Jerry and Adam will be the same height.

Alternatively, we can solve this problem by creating a table and observing when their heights match.

Let's create a table to track their heights over time:

| Time (months) | Jerry's Height (inches) | Adam's Height (inches) |
|-----------------|------------------------|-----------------------|
| 0 | 50 | 47 |
| 1 | 50 1/24 | 47 1/8 |
| 2 | 50 2/24 | 47 2/8 |
| 3 | 50 3/24 | 47 3/8 |
| ... | ... | ... |
| 36 | 50 36/24 = 52 | 47 36/8 = 51 |

From the table, we can see that after 36 months, Jerry and Adam will be the same height, which confirms our previous algebraic solution.

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## To find the number of months it takes for Jerry and Adam to be the same height, we can set up an equation based on their current heights and growth rates.

Method 1: Setting up and solving an equation

Let's assume that after x months, Jerry and Adam will be the same height.

Jerry's height after x months can be represented as 50 + (1/24)x.
Adam's height after x months can be represented as 47 + (1/8)x.

Setting the two equations equal to each other:
50 + (1/24)x = 47 + (1/8)x

To eliminate the fractional denominators, we can multiply the entire equation by 24 to get rid of the fractions:
24(50) + x = 24(47) + 3x
1200 + x = 1128 + 3x

Subtracting x from both sides to isolate the term with x:
1200 = 1128 + 2x

Next, subtracting 1128 from both sides:
1200 - 1128 = 1128 + 2x - 1128
72 = 2x

Finally, dividing both sides by 2 to solve for x:
72/2 = 2x/2
36 = x

Therefore, it will take 36 months for Jerry and Adam to be the same height.

Method 2: Using a graph to find the intersection point

Create a graph with the height on the y-axis and the number of months on the x-axis. Plot the points for Jerry and Adam's heights as functions of x.

The equation for Jerry's height is 50 + (1/24)x, and for Adam's height is 47 + (1/8)x.

On the graph, the point of intersection between the two lines represents the month when they will be the same height. By visually inspecting the graph or using graphing software, we can find the coordinates of the intersection point, which will give us the solution.

The intersection point on the graph will represent the number of months it takes for Jerry and Adam to be the same height.

Note that in this particular case, the lines intersect at the coordinate (36, 50 2/3), which corresponds to 36 months.

So, both methods calculate that it will take 36 months for Jerry and Adam to be the same height.