MATHEMATIC QUIZ

• A train travels at a constant speed of 80 kilometers per hour. It departs from City A at 9:00 AM and arrives at City B at 1:00 PM. How far is City B from City A?

• 320 kilometers

• 340 kilometers

• 230 kilometers

• 403 kilometers

Answer explanation

• Solution: Time = 1:00 PM - 9:00 AM = 4 hours. Distance = Speed × Time = 80 km/h × 4 h = 320 km.

Tags

CCSS.3.MD.A.1

A car travels 120 kilometers in 2 hours and then another 150 kilometers in 3 hours. What is the average speed of the car for the entire journey?

• 54 kilometers per hour.

• 56 kilometers per hour.

• 64 kilometers per hour.

• 45 kilometers per hour.

Answer explanation

• Solution: Total distance = 120 km + 150 km = 270 km. Total time = 2 h + 3 h = 5 h. Average speed = Total distance / Total time = 270 km / 5 h = 54 km/h.

Tags

CCSS.6.RP.A.3B

Tom has a budget of $100 for buying books. If each book costs $8.75, how many books can he buy, and how much money will he have left?

• 51 books, $ 3.76 left.

• 15 books, $3.75 left.

• 11 books, $3.75 left.

• 41 books, $3.65 left.

Answer explanation

• Solution: Number of books = $100 / $8.75 ≈ 11 books. Money left = $100 - (11 × $8.75) = $100 - $96.25 = $3.75.

Tags

CCSS.6.NS.B.3

Lisa wants to buy a new bicycle that costs $300. She saves $15 every week. How many weeks will it take her to save enough money to buy the bicycle?

• It will take Lisa 20 weeks to save enough money to buy the bicycle.

• It will take Lisa 21 weeks to save enough money to buy the bicycle.

• It will take Lisa 30 weeks to save enough money to buy the bicycle.

• It will take Lisa 40 weeks to save enough money to buy the bicycle.

Answer explanation

• Total Cost = $300

• Savings per Week = $15

• Number of Weeks = $300 / $15 = 20 weeks

Tags

CCSS.6.EE.B.7

A recipe for a cake requires 4 cups of flour, 2 cups of sugar, and 1 cup of butter. If you want to make a cake using 10 cups of flour, how many cups of sugar and butter will you need?

• You will need 4 cups of sugar and 5.6 cups of butter.

• You will need 5 cups of sugar and 2.5 cups of butter.

• You will need 8 cups of sugar and 4.6 cups of butter.

• You will need 9 cups of sugar and 2.4 cups of butter.

Answer explanation

• Step 1:

• Find the scaling factor for the recipe.

  • The original amount of flour = 4 cups

  • The new amount of flour = 10 cups

  • Scaling factor = 10 cups / 4 cups = 2.5

• Step 2:

• Use the scaling factor to find the new amounts of sugar and butter.

  • New amount of sugar = Original amount of sugar × Scaling factor

  • New amount of sugar = 2 cups × 2.5 = 5 cups

  • New amount of butter = Original amount of butter × Scaling factor

  • New amount of butter = 1 cup × 2.5 = 2.5 cups

Tags

CCSS.5.MD.A.1

A rectangular park is 60 meters long and 40 meters wide. There is a path of width 2 meters running around the inside of the park. What is the area of the path?

• 
  

• The area of the path is 354 square meters.

• The area of the path is 374 square meters.

• The area of the path is 378 square meters.

• The area of the path is 384 square meters.

Answer explanation

• Step 1:

• Calculate the total area of the park.

  • Length of the park = 60 meters

  • Width of the park = 40 meters

  • Total area = Length × Width = 60 m × 40 m = 2400 m²

• Step 2:

• Calculate the dimensions of the inner rectangle (excluding the path).

  • Width of the path = 2 meters

  • Length of the inner rectangle = 60 meters - 2 × 2 meters = 56 meters

  • Width of the inner rectangle = 40 meters - 2 × 2 meters = 36 meters

  • Area of the inner rectangle = Length × Width = 56 m × 36 m = 2016 m²

• Step 3:

• Calculate the area of the path.

  • Area of the path = Total area - Inner area = 2400 m² - 2016 m² = 384 m²

1. 1. Determine how many units of each product the company should produce to maximize its profit while meeting the customer demand and production time constraints.

2. 2. Calculate the maximum profit the company can achieve.

• The company should produce 7.5 units of Product A and 3 units of Product B to maximize its profit while meeting the production and demand constraints. The maximum profit the company can achieve is $585.

• The company should produce 4.6 units of Product A and 3 units of Product B to maximize its profit while meeting the production and demand constraints. The maximum profit the company can achieve is $249

• The company should produce 46 units of Product A and 3 units of Product B to maximize its profit while meeting the production and demand constraints. The maximum profit the company can achieve is $595.

• The company should produce 8.4 units of Product A and 3 units of Product B to maximize its profit while meeting the production and demand constraints. The maximum profit the company can achieve is $545.

Answer explanation

Step 1:

Define the Variables

Let 𝑥x be the number of units of Product A produced each day. Let 𝑦y be the number of units of Product B produced each day.

Step 2:

Formulate the Constraints

• Production time constraint: 2𝑥+3𝑦≤242x+3y≤24 (since the total production time cannot exceed 24 hours).

• Minimum production constraints:

  • 𝑥≥4x≥4 (at least 4 units of Product A)

  • 𝑦≥3y≥3 (at least 3 units of Product B)

Step 3:

Formulate the Objective Function

The objective is to maximize the profit: Profit=50𝑥+70𝑦Profit=50x+70y

Step 4:

Solve the System of Inequalities

We need to determine the feasible region for the inequalities and find the values of 𝑥x and 𝑦y that maximize the profit within this region.

• Inequality 11: 2𝑥+3𝑦≤242x+3y≤24

• Inequality 22: 𝑥≥4x≥4

• Inequality 33: 𝑦≥3y≥3

First, let's solve for 𝑦y in terms of 𝑥x from the production time constraint: 2𝑥+3𝑦≤242x+3y≤24 3𝑦≤24−2𝑥3y≤24−2x 𝑦≤24−2𝑥3y≤324−2x​

Next, we need to find the intersection points of these constraints on the graph.

Step 5:

Find Intersection Points

1. For 𝑥=4x=4: 𝑦≤24−2(4)3y≤324−2(4)​ 𝑦≤24−83y≤324−8​ 𝑦≤163y≤316​ 𝑦≤5.33y≤5.33

2. For 𝑦=3y=3: 2𝑥+3(3)≤242x+3(3)≤24 2𝑥+9≤242x+9≤24 2𝑥≤152x≤15 𝑥≤7.5x≤7.5

So, the feasible region is bounded by:

• 𝑥≥4x≥4

• 𝑦≥3y≥3

• 2𝑥+3𝑦≤242x+3y≤24

Step 6:

Calculate the Profit at the Corner Points of the Feasible Region

The corner points are where the constraints intersect:

• Point 𝐴(4,3)A(4,3)

• Point 𝐵(7.5,3)B(7.5,3)

• Point 𝐶(4,5.33)C(4,5.33)

1. Profit at Point 𝐴(4,3)A(4,3): Profit=50(4)+70(3)=200+210=410Profit=50(4)+70(3)=200+210=410

2. Profit at Point 𝐵(7.5,3)B(7.5,3): Profit=50(7.5)+70(3)=375+210=585Profit=50(7.5)+70(3)=375+210=585

3. Profit at Point 𝐶(4,5.33)C(4,5.33): Profit=50(4)+70(5.33)=200+373.1=573.1Profit=50(4)+70(5.33)=200+373.1=573.1

Step 7:

Determine the Maximum Profit

The maximum profit occurs at Point 𝐵(7.5,3)B(7.5,3).

• Number of units of Product A to produce: 𝑥=7.5x=7.5

• Number of units of Product B to produce: 𝑦=3y=3

• Maximum Profit: $585

A rectangular garden has a length that is twice its width. Around the garden, there is a path that is 3 meters wide. The total area of the garden including the path is 360 square meters.

1. 1. Determine the dimensions of the garden (length and width).

2. 2. Calculate the area of the garden without the path.

3. 3. Find the area of the path alone.

• 1. 1. The dimensions of the garden are 18 meters in length and 9 meters in width.

  2. 2. The area of the garden without the path is 162 square meters.

  3. 3. The area of the path alone is 198 square meters.

• 1. The dimensions of the garden are 19 meters in length and 10 meters in width.

  2. The area of the garden without the path is 145 square meters.

  3. The area of the path alone is 198 square meters

• 1. The dimensions of the garden are 18 meters in length and 7 meters in width.

  2. The area of the garden without the path is 468 square meters.

  3. The area of the path alone is 198 square meters.

• 1. The dimensions of the garden are 18 meters in length and 9 meters in width.

  2. The area of the garden without the path is 795 square meters.

  3. The area of the path alone is 412 square meters.

Answer explanation

Step 1: Define the Variables

Let 𝑤w be the width of the garden. Then, the length 𝑙l of the garden is 2𝑤2w.

Step 2: Account for the Path

The path is 3 meters wide, so it adds 3 meters to each side of the garden.

• The total width including the path = 𝑤+3+3=𝑤+6w+3+3=w+6.

• The total length including the path = 2𝑤+3+3=2𝑤+62w+3+3=2w+6.

Step 3: Formulate the Area Equation

The total area including the path is given as 360 square meters.

(𝑤+6)(2𝑤+6)=360(w+6)(2w+6)=360

Step 4: Solve the Equation

Expand and simplify the equation:

𝑤⋅2𝑤+𝑤⋅6+6⋅2𝑤+6⋅6=360w⋅2w+w⋅6+6⋅2w+6⋅6=360 2𝑤2+6𝑤+12𝑤+36=3602w2+6w+12w+36=360 2𝑤2+18𝑤+36=3602w2+18w+36=360

Subtract 360 from both sides:

2𝑤2+18𝑤+36−360=02w2+18w+36−360=0 2𝑤2+18𝑤−324=02w2+18w−324=0

Divide by 2 to simplify:

𝑤2+9𝑤−162=0w2+9w−162=0

Solve the quadratic equation using the quadratic formula 𝑤=−𝑏±𝑏2−4𝑎𝑐2𝑎w=2a−b±b2−4ac​​:

𝑎=1,𝑏=9,𝑐=−162a=1,b=9,c=−162

𝑤=−9±92−4⋅1⋅(−162)2⋅1w=2⋅1−9±92−4⋅1⋅(−162)​​ 𝑤=−9±81+6482w=2−9±81+648​​ 𝑤=−9±7292w=2−9±729​​ 𝑤=−9±272w=2−9±27​

This gives us two potential solutions:

𝑤=−9+272=182=9w=2−9+27​=218​=9 𝑤=−9−272=−362=−18w=2−9−27​=2−36​=−18

Since width cannot be negative, we have:

𝑤=9w=9

Step 5: Calculate the Length and Area of the Garden

• Width of the garden: 𝑤=9w=9 meters

• Length of the garden: 𝑙=2𝑤=2×9=18l=2w=2×9=18 meters

Area of the garden without the path:

Area=𝑤×𝑙=9×18=162 square metersArea=w×l=9×18=162 square meters

Step 6: Calculate the Area of the Path Alone

Total area including the path:

Total area=360 square metersTotal area=360 square meters

Area of the garden:

Area of the garden=162 square metersArea of the garden=162 square meters

Area of the path:

Area of the path=Total area−Area of the gardenArea of the path=Total area−Area of the garden Area of the path=360−162=198 square metersArea of the path=360−162=198 square meters

A bookstore sells paperback books for $12 each and hardcover books for $20 each. If a customer buys 3 paperback books and 2 hardcover books, how much does the customer spend in total?

• The customer spends a total of $84.

• The customer spends a total of $75.

• The customer spends a total of $46.

• The customer spends a total of $76.

Answer explanation

Step 1:

Calculate the Cost of Paperback Books

The customer buys 3 paperback books, each costing $12.

Cost of paperback books=3×12=$36Cost of paperback books=3×12=$36

Step 2:

Calculate the Cost of Hardcover Books

The customer buys 2 hardcover books, each costing $20.

Cost of hardcover books=2×20=$40Cost of hardcover books=2×20=$40

Step 3:

Calculate the Total Cost

Add the cost of the paperback books and the hardcover books.

Total cost=$36+$40=$76Total cost=$36+$40=$76

Tags

CCSS.3.OA.D.8

In a class, the ratio of boys to girls is 3:5. If there are 24 boys, how many students are there in total?

• There are 144 students in total.

• There are 174 students in total.

• There are 444 students in total.

• There are 145students in total.

Answer explanation

Step 1:

Determine the Ratio of Boys to Girls

Given that the ratio of boys to girls is 3:5.

Step 2:

Find the Total Number of Girls

Since the ratio of boys to girls is 3:5, let's find the number of girls.

If there are 3 parts representing boys and 5 parts representing girls, then each part represents 1331​ of the total number of students.

Number of girls =5×(number of boys)=5×(number of boys)

=5×24=5×24 =120=120

Step 3:

Find the Total Number of Students

Total number of students =number of boys+number of girls=number of boys+number of girls

=24+120=24+120 =144=144