Ratio and proportions

Question

mrs and mr K decided to share$120 equally between their children for summer vacation. When the eldest got holiday job, he divided his share equally between all his brothers and sisters. If each child was given an extra $4, how many children are there in the K Family?

Please include all working out, thank you in advance..

Answer 1

Hi Natalie, let c represent the number of children in the family. Let x represent how much each gets originally. Then, we can set up an equation for the amount each child gets originally.

 

Equation 1:

120/c = x, amount of money available divided by the number of children equals the amount each child gets.

 

Now, add in the change due to the eldest son getting a job. The number of children sharing the $120 is now c - 1 and each now receives x + 4.

 

Equation 2:

 

120/(c - 1) = x + 4

 

From equation 1, substitute for x:

 

120/(c - 1) = 120/c + 4

 

Clear the fractions by multiplying the entire equation by: c * (c - 1), gives:

 

120 * c = 120 * (c - 1) + 4 * c * (c - 1)

 

Clear the parenthesis:

 

120c = 120c - 120 + 4c² - 4c

 

Cancel 120c from both sides:

 

4c² - 4c - 120 = 0

 

Divide all terms by 4:

 

c² - c - 30 = 0

 

Factor:

 

(c + 5) * (c - 6) = 0

 

Use the zero product rule, a * b = 0, a is equal to zero, b is equal to zero, or both a and b are equal to 0:

 

c + 5 = 0

 

c = -5, we discard this solution since it is not possible to have negative children.

 

c - 6 = 0

 

c = 6

 

There are 6 children in the family.

 

Check:

 

120/c = x

 

120/6 = 20

 

120/(6 - 1) = 20 + 4

 

120/5 = 24

 

24 = 24

 

Values check.

 

Questions?