1. Samuel is 2/3 as old as his brother. In five years, he will be 3/4 as old as his brother will be. Find their present ages.
a. If x = the brother’s age now, write an expression for Samuel’s present age in terms of x.
b. Write an expression for the brother’s age in five years.
c. Write an expression for Samuel’s age in five years.
d. Write an equation that could be used to solve the problem.
e. Solve the equation, showing each step of your work. Find both Samuel’s present age and his brother’s present age.
f. Check your answer.
2. You can mow your whole lawn in 45 minutes, and your neighbor can mow your whole lawn in 30 minutes. How long would it take you to mow the lawn together? Clearly show and explain each step for full credit.
a.Write an equation that could be used to solve this problem.
b. Solve the equation, showing each step of your work. Find the amount of time it would take to mow the lawn together.
c. Check your answer.
1. Let's break down the problem step by step:
a. If x represents the brother's age now, Samuel's present age can be written as (2/3)x. This is because Samuel is 2/3 as old as his brother.
b. The brother's age in five years can be written as x + 5.
c. Samuel's age in five years can be written as (2/3)x + 5.
d. To solve the problem, we can set up an equation using the information given. Since Samuel will be 3/4 as old as his brother in five years, we can write the equation: (2/3)x + 5 = (3/4)(x + 5).
e. Let's solve the equation step by step:
(2/3)x + 5 = (3/4)(x + 5) // Distribute the 3/4 on the right side
(2/3)x + 5 = (3/4)x + (3/4)*5 // Multiply 3/4 by each term inside the parentheses
(2/3)x + 5 = (3/4)x + 15/4 // Combine like terms
To get rid of the fractions, let's multiply both sides of the equation by 12 (the least common denominator of 3 and 4):
12 * [(2/3)x + 5] = 12 * [(3/4)x + 15/4]
8x + 60 = 9x + 45 // Simplify
To isolate the variable, let's subtract 8x from both sides:
8x + 60 - 8x = 9x + 45 - 8x
60 = x + 45 // Simplify further
Now subtract 45 from both sides:
60 - 45 = x + 45 - 45
15 = x
Therefore, the brother's present age (x) is 15 years old.
To find Samuel's present age, substitute x = 15 into the expression (2/3)x:
Samuel's present age = (2/3)(15)
= 30/3
= 10
Samuel is 10 years old, and his brother is 15 years old.
f. To check our answer, let's verify if the given conditions are satisfied. In five years, Samuel will be 10 + 5 = 15 years old, and his brother will be 15 + 5 = 20 years old. According to the information given, Samuel will indeed be 3/4 as old as his brother, verifying that our solution is correct.
2. Let's approach the problem step by step:
a. To solve this problem, we can set up the equation: 1/45 + 1/30 = 1/t, where t represents the time it would take to mow the lawn together.
b. Let's solve the equation step by step:
1/45 + 1/30 = 1/t // Find a common denominator (in this case, it's 90)
(2/90) + (3/90) = 1/t // Convert the fractions to have a common denominator
5/90 = 1/t // Combine like terms
To isolate the variable, let's take the reciprocal of both sides:
1 / (5/90) = 1 / (1/t)
90/5 = t
Simplifying further, we get:
t = 18
Therefore, it would take 18 minutes to mow the lawn together.
c. To check our answer, let's verify if the combined work rate results in mowing the whole lawn in the given time.
In 45 minutes, you can mow 1/45th of the lawn, and in 30 minutes, your neighbor can mow 1/30th of the lawn.
In 18 minutes, the combined work rate would be: 1/45 + 1/30 = 3/90 + 2/90 = 5/90, which is equivalent to 1/18.
Since the combined work rate is equal to mowing the whole lawn (1/18), our solution is correct.
Therefore, it would take 18 minutes to mow the lawn together.