Rectangle Side Length Problem: Area 24 & 42 Sq Mm

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Solving the Rectangle Side Length Problem: Areas of 24 and 42 sq mm

Hey guys! Let's dive into a fun math problem involving rectangles and their areas. We're given two rectangular pieces of paper. One has an area of 24 square millimeters, and the other has an area of 42 square millimeters. The key here is that the side lengths are whole numbers (integers) in millimeters. We stick these two rectangles together by joining their shorter sides. The big question is: what's the length of the longer side of the new rectangle we've created?

Understanding the Rectangle Areas

To start, let's break down what it means for a rectangle to have an area of 24 square millimeters. Remember, the area of a rectangle is calculated by multiplying its length and width (Area = Length × Width). So, we need to find pairs of whole numbers that multiply together to give us 24. These pairs represent the possible side lengths of the first rectangle.

Let's list out the factor pairs of 24:

  • 1 × 24
  • 2 × 12
  • 3 × 8
  • 4 × 6

This means the first rectangle could have dimensions of 1 mm by 24 mm, 2 mm by 12 mm, 3 mm by 8 mm, or 4 mm by 6 mm. Similarly, let’s find the factor pairs for the second rectangle's area, which is 42 square millimeters. We need to find pairs of whole numbers that multiply to 42.

Here are the factor pairs of 42:

  • 1 × 42
  • 2 × 21
  • 3 × 14
  • 6 × 7

So, the second rectangle could have dimensions of 1 mm by 42 mm, 2 mm by 21 mm, 3 mm by 14 mm, or 6 mm by 7 mm. Now we have a good grasp of the possible dimensions for each of our rectangles. Next, we need to figure out how joining them affects the overall dimensions.

Joining the Shorter Sides

The problem states that we're joining the shorter sides of these two rectangles. This is a crucial piece of information! When we join the shorter sides, we're essentially adding them together to form one of the sides of the new, larger rectangle. The longer sides of the original rectangles will then become the length of the new rectangle.

To figure out the possible lengths of the longer side of the new rectangle, we need to consider all the possible combinations of joining the shorter sides. This is where things get interesting because we need to consider different scenarios.

Let's think about this systematically. We'll need to consider each possible shorter side length from the first rectangle (area 24 sq mm) and pair it with each possible shorter side length from the second rectangle (area 42 sq mm). We will add them up and examine the longer side.

Finding the Longer Side: Possible Scenarios

Okay, let's dive into the scenarios. We'll look at each possible pair of rectangles and see what happens when we join their shorter sides.

  • Scenario 1: Rectangle 1 (1x24) and Rectangle 2 (1x42)

    • Shorter sides: 1 mm and 1 mm
    • Joined shorter side length: 1 mm + 1 mm = 2 mm
    • Longer sides: 24 mm and 42 mm
    • Longer side of the new rectangle: 42 mm (since it's the larger of the two)

  • Scenario 2: Rectangle 1 (1x24) and Rectangle 2 (2x21)

    • Shorter sides: 1 mm and 2 mm
    • Joined shorter side length: 1 mm + 2 mm = 3 mm
    • Longer sides: 24 mm and 21 mm
    • Longer side of the new rectangle: 24 mm

  • Scenario 3: Rectangle 1 (1x24) and Rectangle 2 (3x14)

    • Shorter sides: 1 mm and 3 mm
    • Joined shorter side length: 1 mm + 3 mm = 4 mm
    • Longer sides: 24 mm and 14 mm
    • Longer side of the new rectangle: 24 mm

  • Scenario 4: Rectangle 1 (1x24) and Rectangle 2 (6x7)

    • Shorter sides: 1 mm and 6 mm
    • Joined shorter side length: 1 mm + 6 mm = 7 mm
    • Longer sides: 24 mm and 7 mm
    • Longer side of the new rectangle: 24 mm

Let’s continue this for all combinations!

  • Scenario 5: Rectangle 1 (2x12) and Rectangle 2 (1x42)

    • Shorter sides: 2 mm and 1 mm
    • Joined shorter side length: 2 mm + 1 mm = 3 mm
    • Longer sides: 12 mm and 42 mm
    • Longer side of the new rectangle: 42 mm

  • Scenario 6: Rectangle 1 (2x12) and Rectangle 2 (2x21)

    • Shorter sides: 2 mm and 2 mm
    • Joined shorter side length: 2 mm + 2 mm = 4 mm
    • Longer sides: 12 mm and 21 mm
    • Longer side of the new rectangle: 21 mm

  • Scenario 7: Rectangle 1 (2x12) and Rectangle 2 (3x14)

    • Shorter sides: 2 mm and 3 mm
    • Joined shorter side length: 2 mm + 3 mm = 5 mm
    • Longer sides: 12 mm and 14 mm
    • Longer side of the new rectangle: 14 mm

  • Scenario 8: Rectangle 1 (2x12) and Rectangle 2 (6x7)

    • Shorter sides: 2 mm and 6 mm
    • Joined shorter side length: 2 mm + 6 mm = 8 mm
    • Longer sides: 12 mm and 7 mm
    • Longer side of the new rectangle: 12 mm

Let's keep going!

  • Scenario 9: Rectangle 1 (3x8) and Rectangle 2 (1x42)

    • Shorter sides: 3 mm and 1 mm
    • Joined shorter side length: 3 mm + 1 mm = 4 mm
    • Longer sides: 8 mm and 42 mm
    • Longer side of the new rectangle: 42 mm

  • Scenario 10: Rectangle 1 (3x8) and Rectangle 2 (2x21)

    • Shorter sides: 3 mm and 2 mm
    • Joined shorter side length: 3 mm + 2 mm = 5 mm
    • Longer sides: 8 mm and 21 mm
    • Longer side of the new rectangle: 21 mm

  • Scenario 11: Rectangle 1 (3x8) and Rectangle 2 (3x14)

    • Shorter sides: 3 mm and 3 mm
    • Joined shorter side length: 3 mm + 3 mm = 6 mm
    • Longer sides: 8 mm and 14 mm
    • Longer side of the new rectangle: 14 mm

  • Scenario 12: Rectangle 1 (3x8) and Rectangle 2 (6x7)

    • Shorter sides: 3 mm and 6 mm
    • Joined shorter side length: 3 mm + 6 mm = 9 mm
    • Longer sides: 8 mm and 7 mm
    • Longer side of the new rectangle: 8 mm

Almost there! Let's finish the last set of combinations.

  • Scenario 13: Rectangle 1 (4x6) and Rectangle 2 (1x42)

    • Shorter sides: 4 mm and 1 mm
    • Joined shorter side length: 4 mm + 1 mm = 5 mm
    • Longer sides: 6 mm and 42 mm
    • Longer side of the new rectangle: 42 mm

  • Scenario 14: Rectangle 1 (4x6) and Rectangle 2 (2x21)

    • Shorter sides: 4 mm and 2 mm
    • Joined shorter side length: 4 mm + 2 mm = 6 mm
    • Longer sides: 6 mm and 21 mm
    • Longer side of the new rectangle: 21 mm

  • Scenario 15: Rectangle 1 (4x6) and Rectangle 2 (3x14)

    • Shorter sides: 4 mm and 3 mm
    • Joined shorter side length: 4 mm + 3 mm = 7 mm
    • Longer sides: 6 mm and 14 mm
    • Longer side of the new rectangle: 14 mm

  • Scenario 16: Rectangle 1 (4x6) and Rectangle 2 (6x7)

    • Shorter sides: 4 mm and 6 mm
    • Joined shorter side length: 4 mm + 6 mm = 10 mm
    • Longer sides: 6 mm and 7 mm
    • Longer side of the new rectangle: 7 mm

Determining the Possible Lengths

Now, let's gather all the possible lengths of the longer side from our scenarios:

  • 42 mm (appears multiple times)
  • 24 mm (appears multiple times)
  • 21 mm (appears multiple times)
  • 14 mm (appears multiple times)
  • 12 mm
  • 8 mm
  • 7 mm

The Answer: Finding the Possible Values

So, the possible lengths for the longer side of the new rectangle are 7 mm, 8 mm, 12 mm, 14 mm, 21 mm, 24 mm, and 42 mm. Guys, remember to always break down problems step by step, and you'll be able to solve even the trickiest ones! We listed all possible dimensions for both rectangles, considered each pair of short sides being joined, and determined the resulting long side. This thorough approach helps ensure we didn't miss any possibilities. This is how you nail these kinds of questions. Keep practicing, and you'll become a rectangle-solving pro in no time!