At the present time, Mrs. Bee’s age is six years more than four times her son’s age

At the present time, Mrs. Bee’s age is six years more than four times her son’s age. Three years
ago, she was seven times as old as her son was then. If (b) represents Mrs. Bee’s age now and (s)
represents her son’s age now, write a system of equations that could be used to model this
scenario. Use this system of equations to determine, algebraically, the ages of both Mrs. Bee
and her son now. Determine how many years from now Mrs. Bee will be three times as old as
her son will be then.

The Correct Answer and Explanation is :

The correct answer is: 38 years old, and her son is 8 years old. In 7 years

To model the scenario involving Mrs. Bee and her son, we can set up a system of equations based on the information provided. Let ( b ) represent Mrs. Bee’s current age and ( s ) represent her son’s current age.

Setting Up the Equations

1. From the first statement: “Mrs. Bee’s age is six years more than four times her son’s age”:
   [
   b = 4s + 6
   ]
2. From the second statement: “Three years ago, she was seven times as old as her son was then”:
   [
   b – 3 = 7(s – 3)
   ]

Now we have the system of equations:

[
\begin{align} (1) & \quad b = 4s + 6 \ (2) & \quad b – 3 = 7(s – 3) \end{align}
]

Solving the System of Equations

First, let’s simplify equation (2):

[
b – 3 = 7(s – 3) \implies b – 3 = 7s – 21 \implies b = 7s – 18
]

Now we have two expressions for ( b ):

1. ( b = 4s + 6 ) (from equation 1)
2. ( b = 7s – 18 ) (from the simplified equation 2)

Setting them equal to each other:

[
4s + 6 = 7s – 18
]

Rearranging this gives:

[
6 + 18 = 7s – 4s \implies 24 = 3s \implies s = 8
]

Now that we have ( s ), we can find ( b ):

[
b = 4s + 6 = 4(8) + 6 = 32 + 6 = 38
]

Thus, Mrs. Bee’s current age is ( 38 ) years, and her son is ( 8 ) years old.

Future Age Calculation

To determine how many years from now Mrs. Bee will be three times as old as her son, we need to set up the following equation:

Let ( x ) be the number of years from now. Then:

[
b + x = 3(s + x)
]

Substituting the known values of ( b ) and ( s ):

[
38 + x = 3(8 + x) \implies 38 + x = 24 + 3x
]

Rearranging gives:

[
38 – 24 = 3x – x \implies 14 = 2x \implies x = 7
]

Therefore, in 7 years, Mrs. Bee will be three times as old as her son.

Summary

In summary, Mrs. Bee is currently 38 years old, and her son is 8 years old. In 7 years, Mrs. Bee will be three times the age of her son.