1. The sum of two numbers is 176 and their difference is 48. Find the numbers. 2. A 40% sugar solution is added to an 85% sugar solution to create I 800 mL of a 60% solution.How much of each kind of solution is use 3. There were 30000 tickets sold for a baseball game. If only 4 and 10 tickets were sold and the total receipts were 165000 how many tickets of each kind were sold? 4. The perimeter of a rectangle is 46 cm . What are the dimensions if the length is 4 cm less than twice the width?

## Step 1: Setting up equations for the sum and difference<br />### Let the two numbers be $x$ and $y$. We are given that their sum is 176, so $x + y = 176$. Their difference is 48, so $x - y = 48$.<br /><br />## Step 2: Solving for x<br />### Add the two equations: $(x + y) + (x - y) = 176 + 48$. This simplifies to $2x = 224$. Dividing both sides by 2 gives $x = 112$.<br /><br />## Step 3: Solving for y<br />### Substitute the value of $x$ (112) into the first equation: $112 + y = 176$. Subtracting 112 from both sides gives $y = 64$.<br /><br />## Step 4: Solution for problem 1<br />### The two numbers are 112 and 64.<br /><br /><br />## Step 5: Setting up equations for the sugar solutions<br />### Let $x$ be the amount of 40% solution and $y$ be the amount of 85% solution. The total volume is 1800 mL, so $x + y = 1800$. The amount of sugar in the final solution is $0.60 \times 1800 = 1080$ mL. The sugar from each solution adds up to this amount: $0.40x + 0.85y = 1080$.<br /><br />## Step 6: Solving for x and y (sugar solutions)<br />### Multiply the first equation by -0.40: $-0.40x - 0.40y = -720$. Add this to the second equation: $0.45y = 360$. Dividing both sides by 0.45 gives $y = 800$. Substituting this into $x + y = 1800$ gives $x = 1000$.<br /><br />## Step 7: Solution for problem 2<br />### 1000 mL of 40% solution and 800 mL of 85% solution are used.<br /><br /><br />## Step 8: Setting up equations for the tickets<br />### Let $x$ be the number of $4 tickets and $y$ be the number of $10 tickets. The total number of tickets is 30000, so $x + y = 30000$. The total revenue is $165000, so $4x + 10y = 165000$.<br /><br />## Step 9: Solving for x and y (tickets)<br />### Multiply the first equation by -4: $-4x - 4y = -120000$. Add this to the second equation: $6y = 45000$. Dividing both sides by 6 gives $y = 7500$. Substituting this into $x + y = 30000$ gives $x = 22500$.<br /><br />## Step 10: Solution for problem 3<br />### 22500 tickets at $4 and 7500 tickets at $10 were sold.<br /><br /><br />## Step 11: Setting up equations for the rectangle<br />### Let $l$ be the length and $w$ be the width. The perimeter is 46 cm, so $2l + 2w = 46$. The length is 4 cm less than twice the width, so $l = 2w - 4$.<br /><br />## Step 12: Solving for l and w (rectangle)<br />### Substitute the second equation into the first: $2(2w - 4) + 2w = 46$. This simplifies to $6w - 8 = 46$, so $6w = 54$, and $w = 9$. Substituting this into $l = 2w - 4$ gives $l = 2(9) - 4 = 14$.<br /><br />## Step 13: Solution for problem 4<br />### The dimensions are length = 14 cm and width = 9 cm.