Question

Quan's mystery number has 2 tens ? ones. Logan's mystery number has ? tens 5 ones. Logan's number is greater than Quan's number. The difference of the boys' numbers is 42. What are the mystery numbers?

Answer 1

Quan's mystery number is 23 and Logan's mystery number is 65. The equation 10y + 5 = 20 + x + 42 was used to find the values, where 'y' is the tens digit of Logan's number and 'x' is the ones digit of Quan's number. The difference between the two numbers is 42.

Step-by-step explanation:

To solve for Quan's and Logan's mystery numbers, let's use algebraic expressions to represent the unknown values. Begin by letting 'x' represent the digit in the ones place of Quan's number, which would make Quan's number 20 + x. Since Logan's number has 5 ones, if 'y' represents the tens digit in Logan's mystery number, Logan's number would be 10y + 5.

According to the problem, Logan's number is greater than Quan's number by 42. This gives us the equation:

• 10y + 5 = 20 + x + 42

Rewriting the equation, we get:

• 10y - x = 57

Since tens and ones are whole numbers and Logan's number is greater than Quan's, y must be greater than 2. Moreover, we know that the ones digit (x) can only be a single digit (0-9), and it's given that the tens digit for Logan (y) combined with the 5 ones makes the number greater, keeping in mind the difference of 42.

By trying out different values for 'y' and 'x', while maintaining Logan's number greater than Quan's, it becomes clear that when y is 6 (for a number of 65), and x is 3 (for a number of 23), the criteria are met. Thus:

• Quan's number is 23
• Logan's number is 65

And indeed, 65 - 23 = 42.

Answer 2

To find the mystery numbers, set up an inequality and test different values for the variable representing the tens. The mystery numbers are 20 for Quan and 25 for Logan.

Step-by-step explanation:

To find the mystery numbers, let's use variables to represent the unknowns. Let Quan's mystery number be written as 2t and Logan's mystery number as ?t + 5, where t represents tens and ? represents ones. We know that Logan's number is greater than Quan's, so we can set up the inequality ?t + 5 > 2t. Subtracting ?t from both sides gives us 5 > t, which means t can only be 0, 1, 2, 3, or 4. Let's test each value.

If t = 0, Logan's number would be 5, which is not greater than any number with 2 tens. If t = 1, Logan's number would be 15, which is not greater than any number with 2 tens. If t = 2, Logan's number would be 25, which is greater than any number with 2 tens. Therefore, Quan's number is 20 and Logan's number is 25.