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