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The length and breadth of a cardboard are [tex]$2 \frac{1}{5} \, m[tex]$[/tex] and [tex]$[/tex]1 \frac{1}{5} \, m[tex]$[/tex] respectively. The length and breadth of a second cardboard are [tex]$[/tex]3 \frac{1}{5} \, m[tex]$[/tex] and [tex]$[/tex]2 \frac{2}{5} \, m$[/tex] respectively. Both cardboards are divided into 10 equal small pieces.

What is the total area of a cardboard, if it is made using 5 smaller pieces from the first cardboard and 3 smaller pieces from the second cardboard?

Sagot :

GinnyAnswer
Sure, let's break down the problem step-by-step:

### Step 1: Convert mixed fractions to improper fractions

For the first cardboard:
- Length: \( 2 \frac{1}{5} \) converts to an improper fraction:
[tex]\[ 2 \frac{1}{5} = 2 + \frac{1}{5} = \frac{10}{5} + \frac{1}{5} = \frac{11}{5} \][/tex]

- Breadth: \( 1 \frac{1}{5} \) converts to an improper fraction:
[tex]\[ 1 \frac{1}{5} = 1 + \frac{1}{5} = \frac{5}{5} + \frac{1}{5} = \frac{6}{5} \][/tex]

For the second cardboard:
- Length: \( 3 \frac{1}{5} \) converts to an improper fraction:
[tex]\[ 3 \frac{1}{5} = 3 + \frac{1}{5} = \frac{15}{5} + \frac{1}{5} = \frac{16}{5} \][/tex]

- Breadth: \( 2 \frac{2}{5} \) converts to an improper fraction:
[tex]\[ 2 \frac{2}{5} = 2 + \frac{2}{5} = \frac{10}{5} + \frac{2}{5} = \frac{12}{5} \][/tex]

### Step 2: Calculate the area of each cardboard

For the first cardboard:
[tex]\[ \text{Area}_1 = \frac{11}{5} \times \frac{6}{5} = \frac{66}{25} = 2.64 \text{ square meters} \][/tex]

For the second cardboard:
[tex]\[ \text{Area}_2 = \frac{16}{5} \times \frac{12}{5} = \frac{192}{25} = 7.68 \text{ square meters} \][/tex]

### Step 3: Calculate the area of one small piece from each cardboard

Since each cardboard is divided into 10 equal smaller pieces:

For the first cardboard:
[tex]\[ \text{Area of one small piece}_1 = \frac{2.64 \text{ m}^2}{10} = 0.264 \text{ square meters} \][/tex]

For the second cardboard:
[tex]\[ \text{Area of one small piece}_2 = \frac{7.68 \text{ m}^2}{10} = 0.768 \text{ square meters} \][/tex]

### Step 4: Calculate the total area of the new cardboard

The new cardboard is made by using 5 pieces from the first cardboard and 3 pieces from the second cardboard:

[tex]\[ \text{Total area} = (5 \times \text{Area of one small piece}_1) + (3 \times \text{Area of one small piece}_2) \][/tex]

[tex]\[ \text{Total area} = (5 \times 0.264) + (3 \times 0.768) \][/tex]

[tex]\[ \text{Total area} = 1.32 + 2.304 \][/tex]

[tex]\[ \text{Total area} = 3.624 \text{ square meters} \][/tex]

### Final Answer:

The total area of the new cardboard is [tex]\(3.624 \text{ square meters}\)[/tex].
chibabykalu16

Answer:

3.624m^2

Step-by-step explanation:

Given:

  1. First Cardboard

    Length - 2 1/5 m
    Breadth - 1 1/5 m
  2. Second Cardboard

    Length - 3 1/5 m
    Breadth - 2 2/5 m

What is an Area

Area is the amount of space inside the boundary of a two-dimensional flat shape

What is a rectangle

A rectangle is a 4-sided flat shape with straight sides, where all interior angles are right angles (90 degrees) and opposite sides are parallel and of equal length

Solution

  1. Find the area of the first and second cardboard

    Area = Length * Breadth

    First CardBpard
    Area = [tex]2 \frac{1}{5} \times 1 \frac{1}{5} \implies \frac{(5 \times 2) + 1}{5} \times \frac{(5 \times 1) + 1}{5}[/tex]

    Note: The verbal formula is [tex]\frac{(\text{Denominator} \times \text{Whole Number}) + \text{Numerator}}{\text{Denominator}}[/tex]
    Area = [tex]\frac{11}{5} \times \frac{6}{5}[/tex]
    Area = [tex]\frac{66}{25} {\text{m}}^{2}[/tex]

    Second CardBoard
    Area = [tex]3 \frac{1}{5} \times 2 \frac{2}{5} \implies \frac{(5 \times 3) + 1}{5} \times \frac{(5 \times 2) + 2}{5}[/tex]
    Area = [tex]\frac{16}{5} \times \frac{12}{5}[/tex]
    Area = [tex]\frac{192}{25} {\text{m}}^{2}[/tex]

  2. Find one-tenth of the value of both cardboard (first and second)

    [tex]\begin{tabular}{p{0.45\textwidth} p{0.45\textwidth}}\textbf{First CardBoard} & \textbf{Second CardBoard} \\Dividing the fraction by multiplying it with the inverse of 10 & Dividing the fraction by multiplying it with the inverse of 10 \\ \\Area$_2$ = $\frac{66}{25} {\text{m}}^{2} \times \frac{1}{10}$ & Area$_2$ = $\frac{192}{25} {\text{m}}^{2} \times \frac{1}{10}$ \\ \\Area$_2$ = $\frac{66}{250} {\text{m}}^{2}$ & Area$_2$ = $\frac{192}{250} {\text{m}}^{2}$ \\\end{tabular}}[/tex]

    First Cardboard
    - Divide the fraction by multiplying it with the inverse of 10
    - Area2 = (66/25 m²) × (1/10)
    - Area2 = 66/250 m²

    Second Cardboard
    - Divide the fraction by multiplying it with the inverse of 10
    - Area2 = (192/25 m²) × (1/10)
    - Area2 = 192/250 m²

  3. Finding the total area

    Total Area = (Area_2 of first cardboard * 5) + (Area_2 of the second cardboard * 3)
    Total Area = [tex](\frac{66}{250} {\text{m}}^{2} \times 5) + (\frac{192}{250} {\text{m}}^{2} \times 3)[/tex]
    Total Area = [tex](\frac{330}{250} {\text{m}}^{2}) + (\frac{576}{250} {\text{m}}^{2})[/tex]
    Total Area = [tex]\frac{906}{250} {\text{m}}^{2} \rightarrow 3.624 {\text{m}}^{2}[/tex]

Therefore, the total area of the cardboard if made using 5 smaller pieces from the first cardboard and 3 smaller pieces from the second cardboard simplifies to:

[tex]\boxed{3.624m^2 }[/tex]

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