Question

1. the length of a rectangle is 5 less than twice its width. the perimeter is 26 meters. find the dimensions of the rectangle.
2. find two consecutive numbers whose sum is 115.
3. find two consecutive odd numbers whose sum is 144.
4. find two consecutive even numbers whose sum is 126.
5. find three consecutive numbers whose sum is 84.
6. find three consecutive even integers such that the sum of the smallest number and twice the middle number is 20 more than the largest number.

Explanation:

Step1: Solve problem 7

Let the width of the rectangle be $w$. Then the length $l = 2w - 5$. The perimeter formula of a rectangle is $P=2(l + w)$. Substitute $l$ and $P$:
\[

$$\begin{align*} 26&=2((2w - 5)+w)\\ 26&=2(3w - 5)\\ 26&=6w-10\\ 6w&=26 + 10\\ 6w&=36\\ w&=6 \end{align*}$$

\]
If $w = 6$, then $l=2w - 5=2\times6 - 5=7$.

Step2: Solve problem 8

Let the first consecutive number be $x$, then the second is $x + 1$. Their sum is $x+(x + 1)=115$.
\[

$$\begin{align*} x+(x + 1)&=115\\ 2x+1&=115\\ 2x&=115 - 1\\ 2x&=114\\ x&=57 \end{align*}$$

\]
The two consecutive numbers are 57 and 58.

Step3: Solve problem 9

Let the first consecutive odd number be $x$, then the second is $x + 2$. Their sum is $x+(x + 2)=144$.
\[

$$\begin{align*} x+(x + 2)&=144\\ 2x+2&=144\\ 2x&=144 - 2\\ 2x&=142\\ x&=71 \end{align*}$$

\]
The two consecutive odd - numbers are 71 and 73.

Step4: Solve problem 10

Let the first consecutive even number be $x$, then the second is $x + 2$. Their sum is $x+(x + 2)=126$.
\[

$$\begin{align*} x+(x + 2)&=126\\ 2x+2&=126\\ 2x&=126 - 2\\ 2x&=124\\ x&=62 \end{align*}$$

\]
The two consecutive even numbers are 62 and 64.

Step5: Solve problem 11

Let the first consecutive number be $x$, then the second is $x + 1$ and the third is $x + 2$. Their sum is $x+(x + 1)+(x + 2)=84$.
\[

$$\begin{align*} x+(x + 1)+(x + 2)&=84\\ 3x+3&=84\\ 3x&=84 - 3\\ 3x&=81\\ x&=27 \end{align*}$$

\]
The three consecutive numbers are 27, 28 and 29.

Step6: Solve problem 12

Let the first consecutive even integer be $x$, the second be $x + 2$ and the third be $x + 4$. The equation is $x+2(x + 2)=(x + 4)+20$.
\[

$$\begin{align*} x+2x + 4&=x + 4+20\\ 3x+4&=x + 24\\ 3x-x&=24 - 4\\ 2x&=20\\ x&=10 \end{align*}$$

\]
The three consecutive even integers are 10, 12 and 14.

Answer:

1. Width: 6 meters, Length: 7 meters
2. 57 and 58
3. 71 and 73
4. 62 and 64
5. 27, 28 and 29
6. 10, 12 and 14