Mathematical Optimization: Identifying the Exercise with Maximum Result

Q: Why isn't 6 × 15 the highest since 15 is the biggest number?

Great observation! While 15 is the largest individual number, what matters is the product. $ 6 \times 15 = 90 $, but $ 10 \times 10 = 100 $. The most balanced factors often give larger products!

Q: Is there a pattern to predict which will be highest?

Yes! When numbers have the same sum, the pair closest to being equal gives the largest product. Here, all pairs sum to 20, and 10 + 10 is most balanced.

Q: Do I really need to calculate all four products?

Absolutely! Never assume or guess. Calculate each one: 10 × 10 = 1009 × 11 = 998 × 12 = 966 × 15 = 90 This ensures you get the right answer.

Q: What if two products were the same?

If two expressions had equal products, both would be correct answers! But in this problem, each product is different, so only $ 10 \times 10 $ gives the maximum result of 100.

Q: How can I multiply these quickly in my head?

Use these mental math tricks: 10 × 10 is easy (100). For 9 × 11, think (10-1)(10+1) = 100-1 = 99. For 8 × 12, use 8 × 10 + 8 × 2 = 96.

Step-by-step video solution

Watch the teacher solve the problem with clear explanations

00:00 Choose the largest result

00:04 We'll use long multiplication for calculation

00:23 This is the calculation for the first option

00:27 Let's move to the second option

00:31 We'll use long multiplication for calculation

00:36 This is the calculation for the second option

00:39 Let's move to the third option

00:43 We'll use long multiplication for calculation

00:51 This is the calculation for the third option

00:57 Let's move to the fourth option

01:00 We'll use long multiplication for calculation

01:13 This is the calculation for the fourth option

01:21 And this is the solution to the question

Step-by-step written solution

Follow each step carefully to understand the complete solution

1

Understand the problem

Choose the exercise for the highest result

2

Step-by-step solution

To solve this problem, we'll calculate the product of each exercise:

Calculate $10 \times 10 = 100$
Calculate $9 \times 11 = 99$
Calculate $8 \times 12 = 96$
Calculate $6 \times 15 = 90$

Now, let's work through each calculation:

Step 1: Multiply $10 \times 10$ .
This gives us a product of 100.

Step 2: Multiply $9 \times 11$ .
This gives us a product of 99.

Step 3: Multiply $8 \times 12$ .
This gives us a product of 96.

Step 4: Multiply $6 \times 15$ .
This gives us a product of 90.

Upon comparing all calculated products, we find that the highest product is 100. Thus, the exercise that results in this highest product is $10 \times 10$ .

Therefore, the solution to the problem is $10 \times 10$ with a product of 100.

3

Final Answer

$10\times10$

Key Points to Remember

Essential concepts to master this topic

Calculation: Multiply each pair to find the exact product
Technique: Calculate $10 \times 10 = 100$ , then compare all results
Check: Verify 100 > 99 > 96 > 90 to confirm highest ✓

Common Mistakes

Avoid these frequent errors

Guessing without calculating all products
Don't assume the first or largest-looking numbers give the highest result = wrong choice! Students often pick $6 \times 15$ thinking 15 is biggest. Always calculate every single product first.

Practice Quiz

Test your knowledge with interactive questions

FAQ

Everything you need to know about this question

Why isn't 6 × 15 the highest since 15 is the biggest number?

+

Great observation! While 15 is the largest individual number, what matters is the product. $6 \times 15 = 90$ , but $10 \times 10 = 100$ . The most balanced factors often give larger products!

Is there a pattern to predict which will be highest?

+

Yes! When numbers have the same sum, the pair closest to being equal gives the largest product. Here, all pairs sum to 20, and 10 + 10 is most balanced.

Do I really need to calculate all four products?

+

Absolutely! Never assume or guess. Calculate each one:

10 × 10 = 100
9 × 11 = 99
8 × 12 = 96
6 × 15 = 90

This ensures you get the right answer.

What if two products were the same?

+

If two expressions had equal products, both would be correct answers! But in this problem, each product is different, so only $10 \times 10$ gives the maximum result of 100.

How can I multiply these quickly in my head?

+

Use these mental math tricks: 10 × 10 is easy (100). For 9 × 11, think (10-1)(10+1) = 100-1 = 99. For 8 × 12, use 8 × 10 + 8 × 2 = 96.

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