Let's solve the questions from the image you uploaded. I'll transcribe and solve each one.

Question 7:

• Fiona is $x$ years old.
• Thomas is 9 years older than Fiona.
• Cara is twice as old as Fiona.
• The sum of their ages is 51.

(a) Form an equation in terms of $x$:

Let:

• Fiona's age = $x$
• Thomas's age = $x + 9$ (9 years older than Fiona)
• Cara's age = $2x$ (twice Fiona's age)

The sum of their ages is given as 51: $x + (x + 9) + 2x = 51$

Simplifying: $4x + 9 = 51$ $4x = 42$ $x = 10.5$

(b) Solve the equation and work out Fiona's, Thomas's, and Cara's ages:

• Fiona's age = $10.5$ years
• Thomas's age = $10.5 + 9 = 19.5$ years
• Cara's age = $2 \times 10.5 = 21$ years

Question 8:

• Alan is $x$ years old.
• Barry is 10 years younger than Alan.
• Kevin is double Alan's age.
• The sum of their ages is 54.

(a) Form an equation in terms of $x$:

Let:

• Alan's age = $x$
• Barry's age = $x - 10$ (10 years younger than Alan)
• Kevin's age = $2x$ (double Alan's age)

The sum of their ages is given as 54: $x + (x - 10) + 2x = 54$

Simplifying: $4x - 10 = 54$ $4x = 64$ $x = 16$

(b) Solve the equation and work out Alan's, Barry's, and Kevin's ages:

• Alan's age = $16$ years
• Barry's age = $16 - 10 = 6$ years
• Kevin's age = $2 \times 16 = 32$ years

Question 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 $x$:

Let:

• Rebecca's age = $x$
• Mary's age = $x + 8$ (8 years older than Rebecca)
• Jill's age = $3(x + 8)$ (three times older than Mary)

The sum of their ages is given as 67: $x + (x + 8) + 3(x + 8) = 67$

Simplifying: $x + x + 8 + 3x + 24 = 67$ $5x + 32 = 67$ $5x = 35$ $x = 7$

(b) Solve the equation and work out Rebecca's, Mary's, and Jill's ages:

• Rebecca's age = $7$ years
• Mary's age = $7 + 8 = 15$ years
• Jill's age = $3 \times 15 = 45$ years

Would you like more details on any step of the solution?

Here are 5 related questions to further your understanding:

1. How can we use algebra to set up an equation for age-related problems?
2. What happens if the sum of the ages changes in one of the scenarios?
3. How can you check the correctness of your solution once you have solved for $x$?
4. How would the equations change if the age differences were unknown and needed to be calculated?
5. What types of real-world problems can these kinds of age equations help solve?

Tip: When solving word problems, it's helpful to define variables clearly and translate verbal relationships directly into algebraic expressions.