Question 1

Solve.
-Khang and Hector live $40.8$ miles apart in southeastern Missouri. They decide to bicycle towards each other and meet somewhere in between. Hector's rate of speed is $70 \%$ of Khang's. They start out at the same time and meet 4 hours later. Find Hector's rate of speed.

Accepted Answer

The answer of Solve.
-Khang and Hector live $40.8$ miles apart...

Question 2

Solve.
-Two numbers total 8, and their difference is 20. Find the two numbers.

Accepted Answer

The two equations based on the given conditions are x + y = 8 and x - y = 20. Solving these simultaneously, we find x = 14 and y = -6.

Question 3

Solve.
-Doreen and Irena plan to leave their houses at the same time, roller blade towards each other, and meet for lunch after 2 hours on the road. Doreen can maintain a speed of $3.5$ miles per hour, which is $70 \%$ of Irena's speed. If they meet exactly as planned, what is the distance between their houses?

Accepted Answer

The answer of Solve.
-Doreen and Irena plan to leave their...

Question 4

Solve.
-A vendor sells hot dogs and bags of potato chips. A customer buys 5 hot dogs and 3 bags of potato chips for $\$ 17.75$. Another customer buys 2 hot dogs and 2 bags of potato chips for $\$ 8.50$. Find the cost of each item.

Accepted Answer

The answer of Solve.
-A vendor sells hot dogs and bags...

Question 5

Solve.
-Dmitri needs 6 liters of a $16 \%$ solution of sulfuric acid for a research project in molecular biology. He has two supplies of sulfuric acid solution: one is an unlimited supply of the $12 \%$ solution and the other an unlimited supply of the $24 \%$ solution. How many liters of each solution should Dmitri use?

Accepted Answer

This problem can be solved using the method of weighted averages or the concept of mixing solutions. Let $x$ be the amount of $12\%$ solution and $y$ be the amount of $24\%$ solution. We have two equations: $x + y = 6$ (since the total solution needed is 6 liters) and $0.12x + 0.24y = 0.16 \times 6$ (to achieve a $16\%$ solution). Solving these equations simultaneously gives $x = 4$ liters and $y = 2$ liters.

Question 6

Solve.
-Julie and Eric row their boat (at a constant speed) 60 miles downstream for 6 hours, helped by the current. Rowing at the same rate, the trip back against the current takes 10 hours. Find the rate of the current.

Accepted Answer

The speed of the boat in still water is 10 mph (60 miles / 6 hours), and the speed against the current is 6 mph (60 miles / 10 hours). The rate of the current is the difference between these two speeds, which is 2 mph.

Question 7

Solve.
-A barge takes 6 hours to move (at a constant rate) downstream for 60 miles, helped by a current of 2 miles per hour. If the barge's engines are set at the same pace, find the time of its return trip against the current.

Accepted Answer

The barge's speed in still water can be found by subtracting the current's speed from its effective speed downstream. Downstream speed = distance/time = 60 miles / 6 hours = 10 mph. Since the current contributes 2 mph, the barge's speed in still water is 10 mph - 2 mph = 8 mph. Against the current, its effective speed is 8 mph - 2 mph = 6 mph. To travel 60 miles upstream at 6 mph takes 60 miles / 6 mph = 10 hours.

Question 8

Solve.
-Two cars leave a city and head in the same direction. After 5 hours, the faster car is 35 miles ahead of the slower car. The slower car has traveled 245 miles. Find the speeds of the two cars.

Accepted Answer

The slower car travels 245 miles in 5 hours, so its speed is $245 \div 5 = 49 \mathrm{mph}$. The faster car is 35 miles ahead, so it travels $245 + 35 = 280$ miles in 5 hours, making its speed $280 \div 5 = 56 \mathrm{mph}$.

Question 9

Solve.
-Devon purchased tickets to an air show for 9 adults and 2 children. The total cost was $\$ 228$. The cost of a child's ticket was $\$ 7$ less than the cost of an adult's ticket. Find the price of an adult's ticket and a child's ticket.

Accepted Answer

The answer of Solve.
-Devon purchased tickets to an air show...

Question 10

Solve.
-One number is 1 less than a second number. Twice the second number is 10 less than 3 times the first. Find the two numbers.

Accepted Answer

Let the first number be x and the second number be y. The given conditions are: y = x + 1 and 2y = 3x - 10. Substituting y from the first equation into the second gives 2(x + 1) = 3x - 10. Solving this equation for x gives x = 12, and substituting back to find y gives y = 13.

Question 11

Solve.
-Jamil always throws loose change into a pencil holder on his desk and takes it out every two weeks. This time it is all nickels and dimes. There are 7 times as many dimes as nickels, and the value of the dimes is $\$ 7.80$ more than the value of the nickels. How many nickels and dimes does Jamil have?

Accepted Answer

Let the number of nickels be $n$ and the number of dimes be $d$. Given that $d = 7n$ and the value difference is $7.80$, we can set up the equations: $0.10d - 0.05n = 7.80$ and $d = 7n$. Solving these equations gives us $n = 12$ and $d = 84$, which corresponds to choice A.

Question 12

Solve.
-Natasha rides her bike (at a constant speed) for 4 hours, helped by a wind of 2 miles per hour. Pedaling at the same rate, the trip back against the wind takes 8 hours. Find find the total round trip distance she traveled.

Accepted Answer

The wind-assisted speed is (r + 2) mph, and the return speed is (r - 2) mph. The distance one way is 4(r + 2) = 8(r - 2). Solving gives r = 6 mph. The one-way distance is 4(6 + 2) = 32 miles, so the round trip is 64 miles.

Question 13

Solve.
-On a buying trip in Los Angeles, Rosaria Perez ordered 120 pieces of jewelry: a number of bracelets at $\$ 7$ each and a number of necklaces at $\$ 9$ each. She wrote a check for $\$ 1040$ to pay for the order. How many bracelets and how many necklaces did Rosaria purchase?

Accepted Answer

The answer of Solve.
-On a buying trip in Los Angeles,...

Question 14

Solve.
-University Theater sold 513 tickets for a play. Tickets cost $\$ 24$ per adult and $\$ 14$ per senior citizen. If total receipts were $\$ 8482$, how many senior citizen tickets were sold?

Accepted Answer

The answer of Solve.
-University Theater sold 513 tickets for a...

Question 15

Solve.
-Chandra has 5 liters of a $56 \%$ solution of sodium hydroxide in a container. What is the amount and concentration of sodium hydroxide solution she must add to this in order to end up with 7 liters of $46 \%$ solution?

Accepted Answer

To solve this, we can use the equation for the concentration of the final solution. Initially, Chandra has 5 liters of a 56% solution, which means she has $5 \times 0.56 = 2.8$ liters of pure sodium hydroxide. She wants to end up with 7 liters of a 46% solution, which means she needs $7 \times 0.46 = 3.22$ liters of pure sodium hydroxide in total. Therefore, she needs to add $3.22 - 2.8 = 0.42$ liters of pure sodium hydroxide. If she is adding 2 liters of solution to reach 7 liters, the concentration of the solution she needs to add can be found by dividing the amount of pure sodium hydroxide she needs to add by the volume of solution she is adding: $0.42 \div 2 = 0.21$ or 21%. Therefore, she needs to add 2 liters of a 21% sodium hydroxide solution.

Question 16

Solve.
-Jarod is having a problem with rabbits getting into his vegetable garden, so he decides to fence it in. The length of the garden is 4 feet more than 3 times the width. He needs 56 feet of fencing to do the job. Find the length and width of the garden.

Accepted Answer

The answer of Solve.
-Jarod is having a problem with rabbits...

Question 17

Solve.
-The three angles in a triangle always add up to $180 ^ { \circ }$. If one angle in a triangle is $57 ^ { \circ }$ and the second is 2 times the third, what are the three angles?

Accepted Answer

The sum of angles in a triangle is $180^\circ$. Let the third angle be $x$, so the second angle is $2x$. The equation is $57^\circ + 2x + x = 180^\circ$, solving for $x$ gives $x = 41^\circ$, making the second angle $82^\circ$.

Question 18

Solve.
-A certain aircraft can fly 912 miles with the wind in 4 hours and travel the same distance against the wind in 6 hours. What is the speed of the wind?

Accepted Answer

The speed of the aircraft in still air is the average of the speeds with and against the wind. With the wind, the speed is 912 miles / 4 hours = 228 mph. Against the wind, the speed is 912 miles / 6 hours = 152 mph. The average speed in still air is (228 + 152) / 2 = 190 mph. The wind speed is the difference between the speed with the wind and the speed in still air, which is 228 - 190 = 38 mph.

Question 19

Solve.
-One number is four more than a second number. Two times the first number is 8 more than four times the second number.

Accepted Answer

The equations derived from the problem are: 1) x = y + 4, and 2) 2x = 4y + 8. Substituting the first equation into the second gives 2(y + 4) = 4y + 8, which simplifies to y = 0. Substituting y = 0 into the first equation gives x = 4.

Question 20

Solve.
-One number is 6 less than a second number. Twice the second number is 40 more than 4 times the first. Find the two numbers.

Accepted Answer

Let the first number be $x$ and the second number be $y$. The given information translates to two equations: $x = y - 6$ and $2y = 4x + 40$. Substituting the first equation into the second gives $2y = 4(y - 6) + 40$, which simplifies to $2y = 4y - 24 + 40$, leading to $2y = 4y + 16$. Solving for $y$ gives $y = -8$, and substituting back gives $x = -14$.

Quiz 4: Solving Systems of Linear Equations

Solve. -Khang and Hector live $40.8$ miles apart in southeastern Missouri. They decide to bicycle towards each other and meet somewhere in between. Hector's rate of speed is $70 \%$ of Khang's. They start out at the same time and meet 4 hours later. Find Hector's rate of speed.

Solve. -Two numbers total 8, and their difference is 20. Find the two numbers.

Solve. -A vendor sells hot dogs and bags of potato chips. A customer buys 5 hot dogs and 3 bags of potato chips for $\$ 17.75$ . Another customer buys 2 hot dogs and 2 bags of potato chips for $\$ 8.50$ . Find the cost of each item.

Solve. -Julie and Eric row their boat (at a constant speed) 60 miles downstream for 6 hours, helped by the current. Rowing at the same rate, the trip back against the current takes 10 hours. Find the rate of the current.

Solve. -A barge takes 6 hours to move (at a constant rate) downstream for 60 miles, helped by a current of 2 miles per hour. If the barge's engines are set at the same pace, find the time of its return trip against the current.

Solve. -Two cars leave a city and head in the same direction. After 5 hours, the faster car is 35 miles ahead of the slower car. The slower car has traveled 245 miles. Find the speeds of the two cars.

Solve. -Devon purchased tickets to an air show for 9 adults and 2 children. The total cost was $\$ 228$ . The cost of a child's ticket was $\$ 7$ less than the cost of an adult's ticket. Find the price of an adult's ticket and a child's ticket.

Solve. -One number is 1 less than a second number. Twice the second number is 10 less than 3 times the first. Find the two numbers.

Solve. -Natasha rides her bike (at a constant speed) for 4 hours, helped by a wind of 2 miles per hour. Pedaling at the same rate, the trip back against the wind takes 8 hours. Find find the total round trip distance she traveled.

Solve. -On a buying trip in Los Angeles, Rosaria Perez ordered 120 pieces of jewelry: a number of bracelets at $\$ 7$ each and a number of necklaces at $\$ 9$ each. She wrote a check for $\$ 1040$ to pay for the order. How many bracelets and how many necklaces did Rosaria purchase?

Solve. -University Theater sold 513 tickets for a play. Tickets cost $\$ 24$ per adult and $\$ 14$ per senior citizen. If total receipts were $\$ 8482$ , how many senior citizen tickets were sold?

Solve. -Chandra has 5 liters of a $56 \%$ solution of sodium hydroxide in a container. What is the amount and concentration of sodium hydroxide solution she must add to this in order to end up with 7 liters of $46 \%$ solution?

Solve. -Jarod is having a problem with rabbits getting into his vegetable garden, so he decides to fence it in. The length of the garden is 4 feet more than 3 times the width. He needs 56 feet of fencing to do the job. Find the length and width of the garden.

Solve. -The three angles in a triangle always add up to $180 ^ { \circ }$ . If one angle in a triangle is $57 ^ { \circ }$ and the second is 2 times the third, what are the three angles?

Solve. -A certain aircraft can fly 912 miles with the wind in 4 hours and travel the same distance against the wind in 6 hours. What is the speed of the wind?

Solve. -One number is four more than a second number. Two times the first number is 8 more than four times the second number.

Solve. -One number is 6 less than a second number. Twice the second number is 40 more than 4 times the first. Find the two numbers.