Testinar Grade 8 math

Solving Real Problems with Systems

Learn to reason about and solve Grade 8 problems involving Solving Real Problems with Systems.

CH03-06 Equations, Inequalities, and Systems

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Score: ____ / 8

Directions Solve each problem. Show your reasoning on the page or on separate paper. Check the answer key only after finishing.

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?
1. $(10, 30)$
2. $(20, 20)$
3. $(16, 24)$
4. $(30, 10)$
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?
1. $2.5$ hours
2. $3.5$ hours
3. $3$ hours
4. $4$ hours
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?
1. $1$ hour
2. $2$ hours
3. $1.5$ hours
4. $2.5$ hours
Tickets cost $$4$ for children and $$7$ for adults. The theater sold $200$ tickets for a total of $$1{,}100$. How many children's tickets were sold?
1. $100$
2. $150$
3. $50$
4. $120$
A chemist mixes a $30\%$ acid solution with an $80\%$ acid solution to make $100$ mL of a $50\%$ solution. Let $x$ = volume of $30\%$ solution (mL). Which equation represents the acid content?
1. $0.3x + 0.8(100-x) = 0.5(100)$
2. $0.3(100-x) + 0.8x = 50$
3. $30x + 80x = 50 \cdot 100$
4. $0.3x + 0.8x = 50$
Train A travels at $60$ mph. Train B travels at $75$ mph. If Train B leaves $1$ hour later but they meet after Train A has traveled for $5$ hours, which equation is correct?
1. $60 \cdot 5 = 75 \cdot 4$
2. $60 \cdot 5 + 75 \cdot 4 = \text{total distance}$
3. $60 \cdot 4 = 75 \cdot 5$
4. $60 \cdot 5 = 75 \cdot 5$
A piggy bank holds $25$ coins consisting of nickels (5¢) and dimes (10¢). The total value is $$1.70$. How many nickels are in the bank?
1. $10$
2. $16$
3. $12$
4. $8$
Maria is currently $4$ years older than her sister Emma. In $3$ years, Maria will be twice Emma's age. How old is Emma now?
1. $1$
2. $2$
3. $3$
4. $5$

Teacher and parent reference

Answer key and explanations

C. $(16, 24)$
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 $b=24$.
C. $3$ hours
When flying in opposite directions, combined speed $= 480 + 520 = 1000$ mph. Distance $=$ speed $\times$ time: $3000 = 1000t$, so $t = 3$ hours.
C. $1.5$ hours
Boat A's effective upstream speed is $10-2=8$ mph. Boat B's effective downstream speed is $14+2=16$ mph. Their separation increases at $8+16=24$ mph, so $24t=36$ and $t=1.5$ hours.
A. $100$
Let $c=$ children, $a=$ adults. Then $c+a=200$ and $4c+7a=1100$. Substitute $a=200-c$: $4c+7(200-c)=1100$, so $-3c=-300$, $c=100$.
A. $0.3x + 0.8(100-x) = 0.5(100)$
The $30\%$ solution contributes $0.3x$ mL of pure acid. The $80\%$ solution contributes $0.8(100-x)$ mL. Total pure acid must equal $0.5(100) = 50$ mL. Only option A is correct.
A. $60 \cdot 5 = 75 \cdot 4$
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 $60 \cdot 5 = 75 \cdot 4$.
C. $12$
Let $n =$ nickels, $d =$ dimes. Then $n + d = 25$ and $0.05n + 0.10d = 1.70$. From the first equation, $d = 25 - n$. Substituting: $0.05n + 0.10(25-n) = 1.70$, so $0.05n + 2.5 - 0.10n = 1.70$, giving $-0.05n = -0.8$, so $n = 12$.
A. $1$
Let $e =$ Emma's current age. Then Maria's age is $e + 4$. In $3$ years: Emma is $e + 3$, Maria is $e + 7$. The condition gives $e + 7 = 2(e + 3)$, so $e + 7 = 2e + 6$, thus $e = 1$.