n 9: Rebecca is x years old.
Mary is 8 years older than Rebecca.
Jill is three times older than Mary.
The sum of their ages is 67.
(a) Form an equation in terms of [tex]$x$[/tex]
(b) Solve the equation and work out Rebecca's, Mary's, and Jill's ages.
10: Andy has [tex]$x$[/tex] pence.
Kelly has 7 pence more than Andy.
Georgia has 9 pence less than Andy.
The total amount of money they have is [tex]$£ 1.48$[/tex]
Answer (1)
Set up equations based on the relationships between the ages of Rebecca, Mary, and Jill, and the amounts of money Andy, Kelly, and Georgia have.
Solve the equation for Rebecca, Mary, and Jill: x + ( x + 8 ) + 3 ( x + 8 ) = 67 , which simplifies to 5 x + 32 = 67 , and then x = 7 . Therefore, Rebecca is 7, Mary is 15, and Jill is 45.
Solve the equation for Andy, Kelly, and Georgia: x + ( x + 7 ) + ( x − 9 ) = 148 , which simplifies to 3 x − 2 = 148 , and then x = 50 . Therefore, Andy has 50 pence, Kelly has 57 pence, and Georgia has 41 pence.
The ages are Rebecca: 7, Mary: 15, Jill: 45 and the amounts of money are Andy: 50, Kelly: 57, Georgia: 41. R e b ecc a : 7 , M a ry : 15 , J i ll : 45 , A n d y : 50 , Ke ll y : 57 , G eor g ia : 41
Explanation
Problem Analysis Let's break down this word problem step by step. We'll start by defining variables and forming equations based on the given information. Then, we'll solve these equations to find the ages of Rebecca, Mary, and Jill, as well as the amounts of money Andy, Kelly, and Georgia have.
Defining Variables and Equations For Rebecca, Mary, and Jill:
Rebecca's age: x years
Mary's age: x + 8 years
Jill's age: 3 ( x + 8 ) years
Sum of their ages: x + ( x + 8 ) + 3 ( x + 8 ) = 67
For Andy, Kelly, and Georgia:
Andy has x pence.
Kelly has x + 7 pence.
Georgia has x − 9 pence.
Total amount: x + ( x + 7 ) + ( x − 9 ) = 148 (since £1.48 is 148 pence)
Solving for Rebecca, Mary, and Jill's Ages (a) Form an equation in terms of x for Rebecca, Mary and Jill: The equation representing the sum of their ages is: x + ( x + 8 ) + 3 ( x + 8 ) = 67 (b) Solve the equation and work out Rebecca's, Mary's and Jill's ages:
Simplify the equation: x + x + 8 + 3 x + 24 = 67
Combine like terms: 5 x + 32 = 67
Solve for x : 5 x = 67 − 32
5 x = 35
x = 7
Rebecca's age: x = 7 years
Mary's age: x + 8 = 7 + 8 = 15 years
Jill's age: 3 ( x + 8 ) = 3 ( 15 ) = 45 years
Check: 7 + 15 + 45 = 67
Solving for Andy, Kelly, and Georgia's Money Form an equation in terms of x for Andy, Kelly and Georgia: The equation representing the sum of their money is: x + ( x + 7 ) + ( x − 9 ) = 148 Solve the equation and work out Andy's, Kelly's and Georgia's amount of money:
Simplify the equation: x + x + 7 + x − 9 = 148
Combine like terms: 3 x − 2 = 148
Solve for x : 3 x = 148 + 2
3 x = 150
x = 50
Andy has x = 50 pence
Kelly has x + 7 = 50 + 7 = 57 pence
Georgia has x − 9 = 50 − 9 = 41 pence
Check: 50 + 57 + 41 = 148
Final Answer Rebecca is 7 years old, Mary is 15 years old, and Jill is 45 years old. Andy has 50 pence, Kelly has 57 pence, and Georgia has 41 pence.
Examples
Let's say you're planning a family event and need to figure out the ages of different family members to organize activities. By setting up equations like we did, you can easily determine everyone's age based on the relationships between them. Similarly, if you're managing a budget and need to track how much money different people have, you can use equations to keep track of their individual amounts and the total sum. This kind of problem-solving is useful in everyday situations where you need to organize and calculate quantities.
