Grade 8 Solving Real Problems with Systems

Visual Model 1

Question: A juice bar mixes two juices. Juice A is \(25\%\) apple. Juice B is \(75\%\) apple. The shop wants \(40\) liters of a \(55\%\) apple blend. The graph below shows the relationship between volumes of A and B. Which point represents the solution?

• A. \((10, 30)\)
• B. \((20, 20)\)
• C. \((16, 24)\)
• D. \((30, 10)\)

Why it works: Let \(a =\) liters of Juice A and \(b =\) liters of Juice B. Then \(a+b=40\) and \(0.25a+0.75b=0.55(40)=22\). Substitute \(b=40-a\): \(0.25a+0.75(40-a)=22\), so \(30-0.50a=22\). Thus \(a=16\) and \(b=24\).

Answer: \((16, 24)\)

Visual Model 2

Question: Two planes depart from the same airport at the same time, flying in opposite directions. Plane A flies at \(480\) mph; Plane B flies at \(520\) mph. The graph shows their positions over time. After how many hours are the planes \(3000\) miles apart?

• A. \(2.5\) hours
• B. \(3.5\) hours
• C. \(3\) hours
• D. \(4\) hours

Why it works: When flying in opposite directions, combined speed \(= 480 + 520 = 1000\) mph. Distance \(=\) speed \(\times\) time: \(3000 = 1000t\), so \(t = 3\) hours.

Answer: \(3\) hours

Example 1

Question: Two boats depart from the same dock, traveling in opposite directions along a river. Boat A goes upstream at \(10\) mph in still water; Boat B goes downstream at \(14\) mph in still water. The river current is \(2\) mph. After how many hours are they \(36\) miles apart?

• A. \(1\) hour
• B. \(2\) hours
• C. \(1.5\) hours
• D. \(2.5\) hours

1. Boat A's effective upstream speed is \(10-2=8\) mph.
2. Boat B's effective downstream speed is \(14+2=16\) mph.
3. Their separation increases at \(8+16=24\) mph, so \(24t=36\) and \(t=1.5\) hours.

Answer: \(1.5\) hours

Example 2

Question: Tickets cost \($4\) for children and \($7\) for adults. The theater sold \(200\) tickets for a total of \($1{,}100\). How many children's tickets were sold?

• A. \(100\)
• B. \(150\)
• C. \(50\)
• D. \(120\)

1. Let \(c=\) children, \(a=\) adults.
2. Then \(c+a=200\) and \(4c+7a=1100\).
3. Substitute \(a=200-c\): \(4c+7(200-c)=1100\), so \(-3c=-300\), \(c=100\).

Answer: \(100\)

Example 3

Question: A chemist mixes a \(30\%\) acid solution with an \(80\%\) acid solution to make \(100\) mL of a \(50\%\) solution. Let \(x\) = volume of \(30\%\) solution (mL). Which equation represents the acid content?

• A. \(0.3x + 0.8(100-x) = 0.5(100)\)
• B. \(0.3(100-x) + 0.8x = 50\)
• C. \(30x + 80x = 50 \cdot 100\)
• D. \(0.3x + 0.8x = 50\)

1. The \(30\%\) solution contributes \(0.3x\) mL of pure acid.
2. The \(80\%\) solution contributes \(0.8(100-x)\) mL.
3. Total pure acid must equal \(0.5(100) = 50\) mL.
4. Only option A is correct.

Answer: \(0.3x + 0.8(100-x) = 50\)

Problem 1

Question: Train A travels at \(60\) mph. Train B travels at \(75\) mph. If Train B leaves \(1\) hour later but they meet after Train A has traveled for \(5\) hours, which equation is correct?

• A. \(60 \cdot 5 = 75 \cdot 4\)
• B. \(60 \cdot 5 + 75 \cdot 4 = \text{total distance}\)
• C. \(60 \cdot 4 = 75 \cdot 5\)
• D. \(60 \cdot 5 = 75 \cdot 5\)

Why it works: Train A travels for \(5\) hours. Train B leaves \(1\) hour later, so it travels for \(4\) hours. Since they meet after covering the same distance, the equation is \(60 \cdot 5 = 75 \cdot 4\).

Answer: \(60 \cdot 5 = 75 \cdot 4\)

Problem 2

Question: A concert venue sells adult tickets for \($15\) and student tickets for \($8\). If \(120\) total tickets were sold for \($1{,}520\), how many student tickets were sold?

• A. \(40\)
• B. \(60\)
• C. \(80\)
• D. \(100\)

Why it works: Let \(a =\) adult tickets and \(s =\) student tickets. Then \(a+s=120\) and \(15a+8s=1520\). Substitute \(a=120-s\): \(15(120-s)+8s=1520\), so \(1800-7s=1520\). Then \(7s=280\), so \(s=40\).

Answer: \(40\)