1. Let's solve the first problem.
a. We are given that Samuel is 2/3 as old as his brother. Let's assume the brother's age is x. Therefore, Samuel's age can be expressed as (2/3) * x.
b. In five years, the brother's age will be x + 5.
c. In five years, Samuel's age will be (2/3) * x + 5.
d. We can write the equation based on the given information:
(2/3) * x + 5 = (3/4) * (x + 5).
e. Let's solve the equation step by step:
Multiply both sides of the equation by 12 to eliminate the fractions:
12 * ((2/3) * x + 5) = 12 * ((3/4) * (x + 5)).
8x + 60 = 9x + 45. (Note: multiplying both sides by 12 causes the denominators to cancel out).
60 - 45 = 9x - 8x (subtract 8x from both sides)
15 = x.
Therefore, the brother's present age is 15 years.
Now, let's calculate Samuel's present age:
Samuel's age = (2/3) * x = (2/3) * 15 = 10 years.
f. To check our answer, let's substitute x = 15 in the equation:
(2/3) * 15 + 5 = (3/4) * (15 + 5).
10 + 5 = (3/4) * 20.
15 = (3/4) * 20.
Both sides are equal, so our answer is correct.
2. Let's solve the second problem.
a. We are given that you can mow the whole lawn in 45 minutes, and your neighbor can mow the whole lawn in 30 minutes. Let's assume the time it would take for both of you to mow the lawn together is y minutes.
b. To calculate the time it would take to mow the lawn together, we can set up the following equation based on the idea that the amount of work done is equal to the rate multiplied by time:
1/45 + 1/30 = 1/y.
Let's find the common denominator and simplify the equation:
(2/90) + (3/90) = 1/y.
5/90 = 1/y.
1/18 = 1/y.
Therefore, y = 18 minutes.
c. To check our answer, let's substitute y = 18 in the equation:
1/45 + 1/30 = 1/18.
(2/90) + (3/90) = 1/18.
5/90 = 1/18.
Both sides are equal, so our answer is correct.