9. Mrs Low has some money to buy plates and bowls for her new house. If she buys \( p \) plates at \( \$ 12 \) each, she will have \( \$ 3 \) left. If she buys \( (p+4) \) bowls at \( \$ 7 \) each, she will have \( \$ 5 \) left. (i) Find the value of \( p \). (ii) How much money does Mrs Low have to buy plates and bowls? (iii) Mrs Low decides to buy 5 plates. She uses the remaining money to buy bowls. How many bowls can she buy?

To solve the problem, we will set up equations based on the information given. Let \( M \) be the total amount of money Mrs. Low has. ### (i) Find the value of \( p \). From the first condition, if Mrs. Low buys \( p \) plates at \( \$ 12 \) each, she will have \( \$ 3 \) left: \[ M - 12p = 3 \] From this, we can express \( M \): \[ M = 12p + 3 \quad \text{(1)} \] From the second condition, if she buys \( (p + 4) \) bowls at \( \$ 7 \) each, she will have \( \$ 5 \) left: \[ M - 7(p + 4) = 5 \] Expanding this gives: \[ M - 7p - 28 = 5 \] Rearranging this, we find: \[ M = 7p + 33 \quad \text{(2)} \] Now we have two expressions for \( M \) from equations (1) and (2): \[ 12p + 3 = 7p + 33 \] Subtract \( 7p \) from both sides: \[ 5p + 3 = 33 \] Subtract \( 3 \) from both sides: \[ 5p = 30 \] Dividing by \( 5 \): \[ p = 6 \] ### (ii) How much money does Mrs. Low have to buy plates and bowls? Now we can substitute \( p = 6 \) back into either equation (1) or (2) to find \( M \). Using equation (1): \[ M = 12(6) + 3 = 72 + 3 = 75 \] So, Mrs. Low has \( \$ 75 \). ### (iii) Mrs. Low decides to buy 5 plates. How many bowls can she buy? First, we calculate how much money she spends on 5 plates: \[ \text{Cost of 5 plates} = 5 \times 12 = 60 \] Now, we find out how much money she has left after buying the plates: \[ \text{Remaining money} = M - \text{Cost of 5 plates} = 75 - 60 = 15 \] Next, we calculate how many bowls she can buy with the remaining money. Each bowl costs \( \$ 7 \): \[ \text{Number of bowls} = \frac{\text{Remaining money}}{\text{Cost per bowl}} = \frac{15}{7} \approx 2.14 \] Since she can only buy whole bowls, she can buy \( 2 \) bowls. ### Summary of Answers: (i) \( p = 6 \) (ii) Mrs. Low has \( \$ 75 \) (iii) She can buy \( 2 \) bowls.