Numbers That Multiply To 30 And Add To 13: How To Find Them

by Henrik Larsen 60 views

Hey guys! Today, we're diving into a fun mathematical puzzle that involves finding two numbers that satisfy two conditions simultaneously: they must multiply to a specific value (in our case, 30) and add up to another specific value (which is 13). This is a classic problem type often encountered in algebra and number theory, and it's super useful for building your problem-solving skills. So, let's break it down step by step and unlock the secrets to solving these kinds of puzzles.

Understanding the Challenge

The core challenge here is that we aren't just looking for any two numbers. We need a perfect pair that fits both the multiplication and addition criteria. It’s like finding the right key that opens two locks at the same time. This requires a bit of strategic thinking and a systematic approach. Let's first clearly define what we are trying to find. We need two numbers, let’s call them x and y, such that:

  • x * y = 30 (Multiplication Condition)
  • x + y = 13 (Addition Condition)

This might seem daunting at first, but don't worry! We'll explore several methods to tackle this problem effectively.

Method 1: Factor Pairs and Trial-and-Error

One straightforward approach is to list out the factor pairs of the multiplication value (30 in this case) and then check which pair adds up to the addition value (13). A factor pair is simply two numbers that multiply together to give a specific product. So, let’s list the factor pairs of 30:

  • 1 and 30 (1 * 30 = 30)
  • 2 and 15 (2 * 15 = 30)
  • 3 and 10 (3 * 10 = 30)
  • 5 and 6 (5 * 6 = 30)

Now that we have our factor pairs, let's see which pair adds up to 13:

  • 1 + 30 = 31 (No)
  • 2 + 15 = 17 (No)
  • 3 + 10 = 13 (Yes!)
  • 5 + 6 = 11 (No)

Bingo! We found our pair. The numbers 3 and 10 satisfy both conditions: 3 multiplied by 10 equals 30, and 3 plus 10 equals 13. So, the solution is 3 and 10.

This method is quite intuitive and works well for smaller numbers. It helps you understand the relationship between factors and sums, which is fundamental in number theory. However, for larger numbers, listing out all the factor pairs can become quite cumbersome. That's where our next method comes in handy.

Method 2: Algebraic Approach

For a more systematic approach, we can use algebra to solve this problem. Remember our two conditions:

  • x * y = 30
  • x + y = 13

We can use these two equations to solve for x and y. Here's how:

  1. Solve for one variable in terms of the other: From the second equation (x + y = 13), we can solve for x (or y, it doesn't matter which) in terms of y. Let's solve for x: x = 13 - y.
  2. Substitute: Now, we substitute this expression for x into the first equation (x * y = 30): (13 - y) * y = 30.
  3. Expand and Rearrange: Expanding the equation gives us 13y - y² = 30. Rearranging this into a standard quadratic form, we get: y² - 13y + 30 = 0.
  4. Solve the Quadratic Equation: We now have a quadratic equation, which we can solve by factoring, using the quadratic formula, or completing the square. In this case, factoring is the easiest method. We need to find two numbers that multiply to 30 and add up to -13. Those numbers are -3 and -10. So, we can factor the quadratic as: (y - 3)(y - 10) = 0.
  5. Find the Solutions for y: This gives us two possible solutions for y: y = 3 or y = 10.
  6. Find the Solutions for x: Now, we substitute these values of y back into our equation x = 13 - y:
    • If y = 3, then x = 13 - 3 = 10.
    • If y = 10, then x = 13 - 10 = 3.

So, we get the same pair of numbers: 3 and 10. The algebraic method is more structured and can handle more complex problems where the numbers are larger or the relationships are less obvious.

Why This Matters: Real-World Applications and Beyond

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