Question

A rectangle with integer side length has perimeter 10. What is the greatest number of these rectangles that can be cut from a piece of paper with width 24 and length 60?
A. 144
B. 180
C. 240
D. 360
E. 480

Answer 1

Hint: We use initial measurements of rectangle as variables. Write an equation for perimeter using the formula and write all possibilities for the measurements. Calculate area from obtained possibilities. Use a unitary method to calculate number of rectangles that can be cut from piece of paper by dividing the area of piece of paper by area of 1 small rectangle.
 * A rectangle is a quadrilateral having four sides, where opposite sides are equal and parallel. If length of rectangle is ‘l’ and breadth of rectangle is ‘b’ then area of rectangle is \[l \times b\]
* Perimeter of rectangle is \[2(l + b)\]
* Unitary method helps us to calculate the number of units by dividing the total value of multiple units by the value of a single unit.

Complete step-by-step solution:
Let us assume length of the given rectangle be ‘x’ and breadth be ‘y’.
Then we know from the formula of perimeter of rectangle having length ‘l’ and breadth ‘b’ is \[2(l + b)\]
Substitute the value of \[l = x,b = y\]
\[ \Rightarrow \]Perimeter\[ = 2(x + y)\]
Since we are given the value of perimeter as 10,
\[ \Rightarrow 10 = 2(x + y)\]
Cancel same factors from both sides of the equation
\[ \Rightarrow 5 = x + y\]
Shift y to LHS of the equation
\[ \Rightarrow 5 - y = x\]............................… (1)
Now we write possibilities for x and y
If \[y = 1 \Rightarrow x = 4\]
If \[y = 2 \Rightarrow x = 3\]
If \[y = 3 \Rightarrow x = 2\]
If \[y = 4 \Rightarrow x = 1\]
So we have two main possibilities i.e. \[1 \times 4\] and \[2 \times 3\]
Now we are given the piece of paper has length 60 and width 24
\[ \Rightarrow \]Area of piece of paper \[ = \left( {24 \times 60} \right)\]square units
\[ \Rightarrow \]Area of piece of paper \[ = 1440\]square units………………..… (2)
Now we calculate the number of small rectangles that can be cut out from a piece of paper using both the possible dimensions obtained from the perimeter.
Case 1: \[1 \times 4\]
We calculate the area of 1 rectangle of dimension \[1 \times 4\]
\[ \Rightarrow \]Area of 1 rectangle \[ = \left( {1 \times 4} \right)\]square units
\[ \Rightarrow \]Area of piece of paper \[ = 4\]square units
Also, we are given the total area of the piece of paper is 1440 square units.
Now we use a unitary method to calculate the number of rectangles that can be cut from a piece of paper by dividing the area of the piece of paper by an area of 1 small rectangle.
\[ \Rightarrow \]Number of rectangles \[ = \dfrac{{1440}}{4}\]
\[ \Rightarrow \]Number of rectangles \[ = 360\]...................… (3)
Case 1: \[2 \times 3\]
We calculate the area of 1 rectangle of dimension \[2 \times 3\]
\[ \Rightarrow \]Area of 1 rectangle \[ = \left( {2 \times 3} \right)\]square units
\[ \Rightarrow \]Area of piece of paper \[ = 6\]square units
Also, we are given the total area of the piece of paper is 1440 square units.
Now we use a unitary method to calculate the number of rectangles that can be cut from a piece of paper by dividing the area of the piece of paper by an area of 1 small rectangle.
\[ \Rightarrow \]Number of rectangles \[ = \dfrac{{1440}}{6}\]
\[ \Rightarrow \]Number of rectangles \[ = 240\]...................… (4)
Since we need the greatest number of rectangles that can be cut we take the case which has a higher number of rectangles i.e. case 1.
\[\therefore \]Greatest number of rectangles that can be cut from piece of paper is 360

Option D is the correct option.

Note: Many students take all four cases obtained as dimensions from the perimeter of the rectangle, but keep in mind the dimensions are the same in other two cases, they are just interchanged and that does not affect the area or the perimeter of the rectangle.