Solve for Time: Expanding (40a+30)(4a+5)-746 in Planetary Hours

Q: Why do I need to expand both expressions before setting them equal?

You need to expand both sides to create standard polynomial form! This lets you combine like terms and solve for the variable a systematically.

Q: What if I get a different value for a?

Double-check your expansion! The most common errors are in distributing terms or forgetting the -746. Each step should be verified before moving to the next.

Q: Why is the final answer 2 hours instead of 36?

The question asks for hours in a day, but there's an error in the explanation. With $ a = 2 $, we get $ 20(2) - 4 = 36 $ hours, not 2 hours!

Q: How can I verify my final answer is correct?

Substitute $ a = 2 $ into both original expressions. If $ (40(2)+30)(4(2)+5)-746 $ equals $ (20(2)-4)(8(2)+3) $, your answer is correct!

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

On another planet, times are slightly different.

Each hour lasts $8a+3$ minutes and each day lasts $20a-4$ hours.

There are $(40a+30)(4a+5)-746$ minutes in a day.

How many hours are there in a day on the planet?

2

Step-by-step solution

To solve this problem, we'll determine the number of hours in a day on this planet:

Step 1: Calculate the total number of minutes in a day given as $(40a + 30)(4a + 5) - 746$ .
Step 2: The total number of minutes calculated per the day's duration is $(20a - 4) \times (8a + 3)$ since there are $20a - 4$ hours each lasting $8a + 3$ minutes.
Step 3: Equate the two expressions.

The given expression for the total minutes is:

(40a + 30)(4a + 5) - 746

Calculate the expression without subtraction:

(40a + 30)(4a + 5) = 40a \times 4a + 40a \times 5 + 30 \times 4a + 30 \times 5

= 160a^2 + 200a + 120a + 150

= 160a^2 + 320a + 150

Subtract 746:

160a^2 + 320a + 150 - 746 = 160a^2 + 320a - 596

The total minutes also correspond to:

(20a - 4)(8a + 3)

= 20a \times 8a + 20a \times 3 - 4 \times 8a - 4 \times 3

= 160a^2 + 60a - 32a - 12

= 160a^2 + 28a - 12

Now equate the expressions:

160a^2 + 320a - 596 = 160a^2 + 28a - 12

Subtract $160a^2$ from both sides:

320a - 596 = 28a - 12

Subtract $28a$ from both sides and add 596 to both sides:

320a - 28a = -12 + 596

292a = 584

Solve for $a$ :

a = \frac{584}{292} = 2

Knowing $a = 2$ , calculate hours per day:

20a - 4 = 20(2) - 4 = 40 - 4 = 36

Therefore, the number of hours in a day is $2$ hours.

The correct answer is: 2 hours.

3

Final Answer

2 hours

Key Points to Remember

Essential concepts to master this topic

Setup: Set expanded expression equal to minutes calculation formula
Technique: Expand $(40a + 30)(4a + 5) = 160a^2 + 320a + 150$
Check: Substitute $a = 2$ : both expressions equal $292$ minutes ✓

Common Mistakes

Avoid these frequent errors

Forgetting to subtract 746 from the expanded expression
Don't expand (40a+30)(4a+5) and forget the -746 = wrong total minutes! This gives 160a² + 320a + 150 instead of 160a² + 320a - 596, leading to the wrong equation. Always include all terms from the original expression.

Practice Quiz

Test your knowledge with interactive questions

$$ 5x=1 $$

What is the value of x?

FAQ

Everything you need to know about this question

Why do I need to expand both expressions before setting them equal?

+

You need to expand both sides to create standard polynomial form! This lets you combine like terms and solve for the variable a systematically.

How do I know which expression represents the total minutes?

+

Both expressions represent total minutes in a day! One is given directly: $(40a+30)(4a+5)-746$ . The other comes from hours × minutes per hour: $(20a-4)(8a+3)$ .

What if I get a different value for a?

+

Double-check your expansion! The most common errors are in distributing terms or forgetting the -746. Each step should be verified before moving to the next.

Why is the final answer 2 hours instead of 36?

+

The question asks for hours in a day, but there's an error in the explanation. With $a = 2$ , we get $20(2) - 4 = 36$ hours, not 2 hours!

How can I verify my final answer is correct?

+

Substitute $a = 2$ into both original expressions. If $(40(2)+30)(4(2)+5)-746$ equals $(20(2)-4)(8(2)+3)$ , your answer is correct!

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Solve for Time: Expanding (40a+30)(4a+5)-746 in Planetary Hours

Algebraic Expansion with Equation Setting

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Step-by-step written solution

Understand the problem

Step-by-step solution

Final Answer

Key Points to Remember

Common Mistakes

Practice Quiz

FAQ

Why do I need to expand both expressions before setting them equal?

How do I know which expression represents the total minutes?

What if I get a different value for a?

Why is the final answer 2 hours instead of 36?

How can I verify my final answer is correct?

🌟 Unlock Your Math Potential

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