Line of Best Fit

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Question 1 of 17

1. Question
What is the line of best fit for this scatter plot?
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A line of best fit represents most or all of the points of a scatter plot as close as possible.

The line of best fit should be between an equal or close to equal value of points above and below it.

Notice that there are a total of $16$ $16$ points in the plot.

This means that there should be $8$ $8$ points above and below the line at most.

Drawing a line of best fit would be a trial-and-error method.

You can try drawing the line along these three points

If the line was drawn along these three points, there will be $6$ $6$ points above the line and $7$ $7$ points below it.

Finally, connect the points with a line.
Question 2 of 17

2. Question
What is the line of best fit for this scatter plot?
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A line of best fit represents most or all of the points of a scatter plot as close as possible.

The line of best fit should be between an equal or close to equal value of points above and below it.

Notice that there are a total of $29$ $29$ points in the plot.

This means that there should be $14$ $14$ to $15$ $15$ points above and below the line at most.

Drawing a line of best fit would be a trial-and-error method.

You can try drawing the line along these two points

If the line was drawn along these two points, there will be $14$ $14$ points above the line and $13$ $13$ points below it.

Finally, connect the points with a line.
Question 3 of 17

3. Question
What is the line of best fit for this scatter plot?
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- 2.
- 3.
- 4.
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A line of best fit represents most or all of the points of a scatter plot as close as possible.

The line of best fit should be between an equal or close to equal value of points above and below it.

Notice that there are a total of $15$ $15$ points in the plot.

This means that there should be $7$ $7$ to $8$ $8$ points above and below the line at most.

Drawing a line of best fit would be a trial-and-error method.

You can try drawing the line from this point down to the lower right end of the plot

If the line was drawn from this point, there will be $7$ $7$ points above the line and $7$ $7$ points below it.

Finally, draw a line from the top point down to the lower right corner of the plot.
Question 4 of 17

4. Question
What is the line of best fit for this scatter plot?
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A line of best fit represents most or all of the points of a scatter plot as close as possible.

The line of best fit should be between an equal or close to equal value of points above and below it.

Notice that there are a total of $28$ $28$ points in the plot.

This means that there should be $14$ $14$ points above and below the line at most.

Drawing a line of best fit would be a trial-and-error method.

You can try drawing the line along these two points

If the line was drawn along these two points, there will be $13$ $13$ points above the line and $13$ $13$ points below it.

Finally, connect the points with a line.
Question 5 of 17

5. Question
Given that Zac is $160 cm$ $160 \text(cm)$ and he is $20 cm$ $20 \text(cm)$ shorter than Tony, find the shoe size of Tony
- (11)
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A line of best fit represents most or all of the points of a scatter plot as close as possible.

First, find the height of Tony.

$Tony’s Height$ $\text(Tony’s Height)$ $=$ $=$ $Zac’s Height + 20$ $\text(Zac’s Height)+20$

$=$ $=$ $160 + 20$ $160+20$

$=$ $=$ $180$ $180$

Finally, use the scatter plot to find the correlation coefficient of Tony’s height.

The line of best fit indicates the best estimate of the correlation coefficient.

$11$ $11$

Question 6 of 17

A group of

17

$17$ students got their shoe sizes and heights measured and the data was listed in a scatter plot. Find the equation of the line of best fit.

$S =$ $S=$ (0.35) $H$ $H$ (-51)

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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

y =

$y=$ $m$ $m$

x +

$x+$ $b$ $b$

First, use the gradient formula to find the gradient or slope of the line.

Pick two points to use for the formula

(

$($ $170$ $170$

,

$,$ $8.5$ $8.5$

) = (

$)=($ $x_{1}$ $x_1$

,

$,$ $y_{1}$ $y_1$

)

$)$

(

$($ $180$ $180$

,

$,$ $12$ $12$

) = (

$)=($ $x_{2}$ $x_2$

,

$,$ $y_{2}$ $y_2$

)

$)$

$m$ $m$	$=$ $=$	$\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$	Gradient Formula

	$=$ $=$	$\frac{\color{#9a00c7}{12-8.5}}{\color{#00880A}{180-170}}$	Substitute values

	$=$ $=$	$\frac{3.5}{10}$

$m$ $m$	$=$ $=$	$0.35$

Next, find the

y

$y$ intercept.

Note that

y = S

$y=S$ (Shoe Size) and

x = H

$x=H$ (Height)

Substitute known values to the gradient-intercept form and solve for $b$ $b$

Use

x_{2}

$x_2$ and

y_{2}

$y_2$ as substitutes for

x

$x$ and

y

$y$ .

$y$ $y$	$=$ $=$	$m$ $m$ $x +$ $x+$ $b$ $b$	Gradient-Intercept Form

$12$ $12$	$=$ $=$	$0.35$ $0.35$ $(180) +$ $(180)+$ $b$ $b$	Substitute values

$12$ $12$ $- 63$ $-63$	$=$ $=$	$63 + b$ $63+b$ $- 63$ $-63$	Subtract $63$ $63$ from both sides
$- 51$ $-51$	$=$ $=$	$b$ $b$
$b$ $b$	$=$ $=$	$- 51$ $-51$

Finally, substitute the values into the gradient-intercept form

m = 0.35

$m=0.35$

b = - 51

$b=-51$

$y$ $y$	$=$ $=$	$m$ $m$ $x +$ $x+$ $b$ $b$	Gradient-Intercept Form

$y$ $y$	$=$ $=$	$0.35$ $0.35$ $x -$ $x-$ $51$ $51$	Substitute values

$S$ $S$	$=$ $=$	$0.35 H - 51$ $0.35H-51$	Change $y$ $y$ and $x$ $x$ to $S$ $S$ and $H$ $H$ respectively

S = 0.35 H - 51

$S=0.35H-51$

Question 7 of 17

7. Question
Using the scatter plot below, find the difference in length of the right forearm of a right foot measuring $21 cm$ $21 \text(cm)$ and a right foot measuring $25 cm$ $25 \text(cm)$ .
- 1. $3 \frac{1}{2}$ $3 1/2$
- 2. $4 \frac{1}{2}$ $4 1/2$
- 3. $4$ $4$
- 4. $5$ $5$
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A line of best fit represents most or all of the points of a scatter plot as close as possible.

First, use the scatter plot to find the correlation coefficient of the given measurements for the right foot.

The line of best fit indicates the best estimate of the correlation coefficient.

$21$ $21$ $:$ $:$ $24$ $24$

$25$ $25$ $:$ $:$ $27 \frac{1}{2}$ $27 1/2$

Finally, find the difference of the right forearms’ measurements.

$27 \frac{1}{2} - 24$ $27 1/2-24$ $=$ $=$ $3 \frac{1}{2}$ $3 1/2$

$3 \frac{1}{2}$ $3 1/2$

Question 8 of 17

A group of

26

$26$ students took a Probability test and then later a Trigonometry test. Their scores were tabulated in a scatter plot as shown below.

(i)

$(i)$ How many students scored more than

70 %

$70%$ in Trigonometry?

(i i)

$(ii)$ What is the equation of the line of best fit?

Write fractions in the format “a/b”

$(i)$ $(i)$ (10) students

$(i i) P =$ $(ii) P=$ (2/3) $T +$ $T+$ (20)

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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

m = \frac{y_{2} - y_{1}}{x_{2} - x_{1}}

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

y =

$y=$ $m$ $m$

x +

$x+$ $b$ $b$

(i)

$(i)$ Find how many students scored more than

70 %

$70%$ in Trigonometry.

Place a ruler on 70 along the Trigonometry axis and simply count the dots on the right side.

There are

10

$10$ dots on the right.

This means there are

10

$10$ students who scored more than

70 %

$70%$ in Trigonometry.

(i i)

$(ii)$ Find the equation of the line of best fit.

First, use the gradient formula to find the gradient or slope of the line.

Pick two points to use for the formula

(

$($ $30$ $30$

,

$,$ $40$ $40$

) = (

$)=($ $x_{1}$ $x_1$

,

$,$ $y_{1}$ $y_1$

)

$)$

(

$($ $90$ $90$

,

$,$ $80$ $80$

) = (

$)=($ $x_{2}$ $x_2$

,

$,$ $y_{2}$ $y_2$

)

$)$

$m$ $m$	$=$ $=$	$\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$	Gradient Formula

	$=$ $=$	$\frac{\color{#9a00c7}{80-40}}{\color{#00880A}{90-30}}$	Substitute values

	$=$	$\frac{40}{60}$

$m$	$=$	$\frac{2}{3}$

Next, find the

$y$ intercept.

Note that

$y=P$ (Probability) and

$x=T$ (Trigonometry)

The line touches the

$y$ axis at

$20$ , hence $b=20$

Finally, substitute the values into the gradient-intercept form

$m=2/3$

$b=20$

$y$	$=$	$m$ $x+$ $b$	Gradient-Intercept Form

$y$	$=$	$2/3$ $x+$ $20$	Substitute values

$P$	$=$	$2/3T+20$	Change $y$ and $x$ to $P$ and $T$ respectively

$(i) 10$ students

$(ii) P=2/3T+20$

Question 9 of 17

The age and height correlation of girls aged

$11$ to

$18$ are listed within a scatter plot. Find the equation of the line of best fit.

$H=$ (8) $A+$ (42)

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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

$y=$ $m$

$x+$ $b$

First, use the gradient formula to find the gradient or slope of the line.

Pick two points to use for the formula

$($ $11$

$,$ $130$

$)=($ $x_1$

$,$ $y_1$

$)$

$($ $16$

$,$ $170$

$)=($ $x_2$

$,$ $y_2$

$)$

$m$	$=$	$\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$	Gradient Formula

	$=$	$\frac{\color{#9a00c7}{170-130}}{\color{#00880A}{16-11}}$	Substitute values

	$=$	$\frac{40}{5}$

$m$	$=$	$8$

Next, find the

$y$ intercept.

Note that

$y=H$ (Height) and

$x=A$ (Age)

Substitute known values to the gradient-intercept form and solve for $b$

Use

$x_2$ and

$y_2$ as substitutes for

$x$ and

$y$ .

$y$	$=$	$m$ $x+$ $b$	Gradient-Intercept Form

$170$	$=$	$8$ $(16)+$ $b$	Substitute values

$170$ $-128$	$=$	$128+b$ $-128$	Subtract $128$ from both sides
$42$	$=$	$b$
$b$	$=$	$42$

Finally, substitute the values into the gradient-intercept form

$m=8$

$b=42$

$y$	$=$	$m$ $x+$ $b$	Gradient-Intercept Form

$y$	$=$	$8$ $x+$ $42$	Substitute values

$H$	$=$	$8A+42$	Change $y$ and $x$ to $H$ and $A$ respectively

$H=8A+42$

Question 10 of 17

10. Question
The age and height correlation of girls aged $11$ to $18$ are listed within a scatter plot. Given that the equation of the line of best fit is $H=8A+42$ , solve for the height of an $18$ year old girl.
- $H=$ (186) $\text(cm)$
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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

$y=$ $m$ $x+$ $b$

Recall that $y=H$ (Height) and $x=A$ (Age)

Since we are given the age $(A)$ of $18$ years old, we can substitute that value to the line of best fit’s formula and solve for the height $(H)$ .

$H$ $=$ $8A+42$

$H$ $=$ $8(18)+42$ Substitute $A=18$

$H$ $=$ $144+42$ Evaluate

$H$ $=$ $186$

$H=186 \text(cm)$

Question 11 of 17

A scatter plot is made to show the correlation of the height of a spring as a weight is attached to it. Find the value of the gradient.

Round your answer to one decimal place

$m=$ (2.1)

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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

$y=$ $m$

$x+$ $b$

First, use the gradient formula to find the gradient or slope of the line.

Pick two points to use for the formula

$($ $0$

$,$ $18$

$)=($ $x_1$

$,$ $y_1$

$)$

$($ $35$

$,$ $90$

$)=($ $x_2$

$,$ $y_2$

$)$

$m$	$=$	$\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$	Gradient Formula

	$=$	$\frac{\color{#9a00c7}{90-18}}{\color{#00880A}{35-0}}$	Substitute values

	$=$	$\frac{72}{35}$

$m$	$=$	$2.1$	Rounded to one decimal place

$m=2.1$

Question 12 of 17

12. Question
A scatter plot is made to show the correlation of the height of a spring as a weight is attached to it. Given that the gradient is $2.1,$ find the equation of the line of best fit.
- $L=$ (2.1) $W+$ (18)
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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

$y=$ $m$ $x+$ $b$

First, find the $y$ intercept.

Note that $y=L$ (Length) and $x=W$ (Weight)

The line touches the $y$ axis at $18$ , hence $b=18$

Finally, substitute the values into the gradient-intercept form

Recall that the gradient of this scatter plot is $m=2.1$ .

$m=2.1$

$b=18$

$y$ $=$ $m$ $x+$ $b$ Gradient-Intercept Form

$y$ $=$ $2.1$ $x+$ $18$ Substitute values

$L$ $=$ $2.1W+18$ Change $y$ and $x$ to $P$ and $T$ respectively

$L=2.1W+18$

Question 13 of 17

$5$ trees were planted and their height per year was recorded in a scatter plot. Find the equation of the line of best fit.

Write fractions in the format “a/b”

$H=$ (5/4) $N+$ (3)

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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

$y=$ $m$

$x+$ $b$

First, use the gradient formula to find the gradient or slope of the line.

Pick two points to use for the formula

$($ $0$

$,$ $3$

$)=($ $x_1$

$,$ $y_1$

$)$

$($ $4$

$,$ $8$

$)=($ $x_2$

$,$ $y_2$

$)$

$m$	$=$	$\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$	Gradient Formula

	$=$	$\frac{\color{#9a00c7}{8-3}}{\color{#00880A}{4-0}}$	Substitute values

	$=$	$\frac{5}{4}$

$m$	$=$	$5/4$

Next, find the

$y$ intercept.

Note that

$y=H$ (Height) and

$x=N$ (Years)

The line touches the

$y$ axis at

$3$ , hence $b=3$

Finally, substitute the values into the gradient-intercept form

$m=5/4$

$b=3$

$y$	$=$	$m$ $x+$ $b$	Gradient-Intercept Form

$y$	$=$	$5/4$ $x+$ $3$	Substitute values

$H$	$=$	$5/4N+3$	Change $y$ and $x$ to $H$ and $N$ respectively

$H=5/4N+3$

Question 14 of 17

14. Question
$5$ trees were planted and their height per year was recorded in a scatter plot. Given that $H=5/4N+3$ , find the height of the trees after $8$ years.
- $H=$ (13) $\text(m)$
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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

$y=$ $m$ $x+$ $b$

Recall that $y=H$ (Height) and $x=N$ (Years)

Since we are given $8$ years $(N)$ , we can substitute that value to the line of best fit’s formula and solve for the height $(H)$ .

$H$ $=$ $5/4N+3$

$H$ $=$ $5/4(8)+3$ Substitute $N=8$

$H$ $=$ $40/4+3$ Evaluate

$H$ $=$ $10+3$

$H$ $=$ $13$

$H=13 \text(m)$

Question 15 of 17

The gain of weight for a group of dolphins was recorded in a scatter plot for

$5$ months. Find the equation of the line of best fit.

Write fractions in the format “a/b”

$W=$ (7/5) $N+$ (1)

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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

$y=$ $m$

$x+$ $b$

First, use the gradient formula to find the gradient or slope of the line.

Pick two points to use for the formula

$($ $0$

$,$ $1$

$)=($ $x_1$

$,$ $y_1$

$)$

$($ $5$

$,$ $8$

$)=($ $x_2$

$,$ $y_2$

$)$

$m$	$=$	$\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$	Gradient Formula

	$=$	$\frac{\color{#9a00c7}{8-1}}{\color{#00880A}{5-0}}$	Substitute values

	$=$	$\frac{7}{5}$

$m$	$=$	$7/5$

Next, find the

$y$ intercept.

Note that

$y=W$ (Weight) and

$x=N$ (Months)

The line touches the

$y$ axis at

$1$ , hence $b=1$

Finally, substitute the values into the gradient-intercept form

$m=7/5$

$b=1$

$y$	$=$	$m$ $x+$ $b$	Gradient-Intercept Form

$y$	$=$	$7/5$ $x+$ $1$	Substitute values

$W$	$=$	$7/5N+1$	Change $y$ and $x$ to $W$ and $N$ respectively

$W=7/5N+1$

Question 16 of 17

16. Question
The gain of weight for a group of dolphins was recorded in a scatter plot for $5$ months. Given that $W=7/5N+1$ , find the weight of the dolphins after $10$ months.
- $W=$ (25) $\text(kg)$
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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

$y=$ $m$ $x+$ $b$

Recall that $y=W$ (Weight) and $x=N$ (Months)

Since we are given $10$ months $(N)$ , we can substitute that value to the line of best fit’s formula and solve for the Weight $(W)$ .

$W$ $=$ $7/5N+1$

$W$ $=$ $7/5(10)+1$ Substitute $N=10$

$W$ $=$ $70/5+1$ Evaluate

$W$ $=$ $24+1$

$W$ $=$ $25$

$W=25 \text(kg)$

Question 17 of 17

Stuart collected data on the height of a lit candle and the time it burns. He created a scatter plot for it and constructed the line of best fit as shown below. Find the equation for the line.

$H=$ (-3) $N+$ (30)

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The Equation of the Line of Best Fit is an equation that best represents a certain correlation between two groups of data.

Gradient Formula

$\color{#007DDC}{m}=\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$

Gradient-Intercept Form

$y=$ $m$

$x+$ $b$

First, use the gradient formula to find the gradient or slope of the line.

Pick two points to use for the formula

$($ $0$

$,$ $30$

$)=($ $x_1$

$,$ $y_1$

$)$

$($ $10$

$,$ $0$

$)=($ $x_2$

$,$ $y_2$

$)$

$m$	$=$	$\frac{\color{#9a00c7}{y_2-y_1}}{\color{#00880A}{x_2-x_1}}$	Gradient Formula

	$=$	$\frac{\color{#9a00c7}{0-30}}{\color{#00880A}{10-0}}$	Substitute values

	$=$	$\frac{-30}{10}$

$m$	$=$	$-3$

Next, find the

$y$ intercept.

Note that

$y=H$ (Height) and

$x=N$ (Hours)

The line touches the

$y$ axis at

$30$ , hence $b=30$

Finally, substitute the values into the gradient-intercept form

$m=-3$

$b=30$

$y$	$=$	$m$ $x+$ $b$	Gradient-Intercept Form
$y$	$=$	$-3$ $x+$ $30$	Substitute values

$H$	$=$	$-3N+30$	Change $y$ and $x$ to $H$ and $N$ respectively

$H=-3N+30$

$Tony’s Height$ $\text(Tony’s Height)$	$=$ $=$	$Zac’s Height + 20$ $\text(Zac’s Height)+20$
	$=$ $=$	$160 + 20$ $160+20$
	$=$ $=$	$180$ $180$

$H$	$=$	$8A+42$

$H$	$=$	$8(18)+42$	Substitute $A=18$

$H$	$=$	$144+42$	Evaluate
$H$	$=$	$186$

$y$	$=$	$m$ $x+$ $b$	Gradient-Intercept Form

$y$	$=$	$2.1$ $x+$ $18$	Substitute values

$L$	$=$	$2.1W+18$	Change $y$ and $x$ to $P$ and $T$ respectively

$H$	$=$	$5/4N+3$

$H$	$=$	$5/4(8)+3$	Substitute $N=8$

$H$	$=$	$40/4+3$	Evaluate

$H$	$=$	$10+3$
$H$	$=$	$13$

$W$	$=$	$7/5N+1$

$W$	$=$	$7/5(10)+1$	Substitute $N=10$

$W$	$=$	$70/5+1$	Evaluate

$W$	$=$	$24+1$
$W$	$=$	$25$

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