Variables, formulas and substitution

1. A museum charges 6 dollars per ticket plus one booking fee of 4 dollars. The formula is $C=6n+4$, where $C$ is the total cost in dollars. State what $n$ represents, give the coefficient of $n$, and explain what the constant term represents.

   3 marks
2. Rewrite $5a$, $ab$ and $3\left(b+2\right)$ using multiplication signs. Then evaluate $ab$ when $a=2$ and $b=4$.

   4 marks
3. When $u=3$ and $v=5$, evaluate

   1. (a)$3u+v$ and
   2. (b)$u+u+v$. Show each substitution. Are the results equal?

   4 marks
4. Evaluate $4\left(p+2\right)$ when $p=3$. Leo instead calculates $4\times 3+2=14$. Explain why his method does not match the expression.

   3 marks
5. Evaluate $s^2+2s$ when $s=5$. Show the values of $s^2$ and $2s$ separately before adding them.

   3 marks
6. Three seedlings have heights $a$, $b$ and $c$, all in centimetres. Their mean height is given by $m=\frac{a+b+c}{3}$. Find $m$ when $a=12$, $b=15$ and $c=18$. Explain why you add before dividing.

   3 marks
7. A rectangular label has length $l=9$ cm and width $w=4$ cm. Use $P=2l+2w$ to find its perimeter. State the unit and check your answer by adding its four side lengths.

   3 marks
8. A bike hire business uses $C=7h+4$, where $C$ is the total cost in dollars and $h$ is the number of hours. Find the cost for

   1. (a)2 hours and
   2. (b)4 hours. Mia says doubling 2 hours to 4 hours doubles the total cost. Use your results and the formula to explain whether she is correct.

   4 marks

Equivalent correct methods earn the corresponding marks. Include units where asked.

1. A museum charges 6 dollars per ticket plus one booking fee of 4 dollars. The formula is $C=6n+4$, where $C$ is the total cost in dollars. State what $n$ represents, give the coefficient of $n$, and explain what the constant term represents.

   3 marks

   The repeated charge is 6 dollars for each ticket, so $n$ counts tickets. (1 mark)

   The number multiplying $n$ is its coefficient, $6$. (1 mark)

   The term $4$ adds one booking fee regardless of the number of tickets. (1 mark)

   $n$ is the number of tickets; its coefficient is $6$; the constant $4$ is the single booking fee in dollars.
2. Rewrite $5a$, $ab$ and $3\left(b+2\right)$ using multiplication signs. Then evaluate $ab$ when $a=2$ and $b=4$.

   4 marks

   $5a=5\times a$. (1 mark)

   $ab=a\times b$. (1 mark)

   $3\left(b+2\right)=3\times \left(b+2\right)$. Keep the whole bracket together. (1 mark)

   Substitute both values: $ab=2\times 4=8$. (1 mark)

   $5\times a$, $a\times b$, $3\times \left(b+2\right)$; $ab=8$.
3. When $u=3$ and $v=5$, evaluate

   1. (a)$3u+v$ and
   2. (b)$u+u+v$. Show each substitution. Are the results equal?

   4 marks

   (a) Replace both variables: $3u+v=3\times 3+5$. (1 mark)

   Calculate $9+5=14$. (1 mark)

   (b) Replace every occurrence: $u+u+v=3+3+5$. (1 mark)

   Calculate $11$. Since $14\ne 11$, the results are different. (1 mark)

   1. (a)$14$;
   2. (b)$11$. The results are different.
4. Evaluate $4\left(p+2\right)$ when $p=3$. Leo instead calculates $4\times 3+2=14$. Explain why his method does not match the expression.

   3 marks

   Substitute while retaining brackets: $4\left(p+2\right)=4\left(3+2\right)$. (1 mark)

   Calculate the bracket, then multiply: $4\times 5=20$. (1 mark)

   The 4 multiplies the whole sum $p+2$. Leo's calculation multiplies only $p$ by 4. (1 mark)

   $20$. Leo has not multiplied the whole bracket by 4.
5. Evaluate $s^2+2s$ when $s=5$. Show the values of $s^2$ and $2s$ separately before adding them.

   3 marks

   Square the value: $s^2=5^2=5\times 5=25$. (1 mark)

   Double the value: $2s=2\times 5=10$. (1 mark)

   Add: $s^2+2s=25+10=35$. (1 mark)

   $s^2=25$, $2s=10$, so $s^2+2s=35$.
6. Three seedlings have heights $a$, $b$ and $c$, all in centimetres. Their mean height is given by $m=\frac{a+b+c}{3}$. Find $m$ when $a=12$, $b=15$ and $c=18$. Explain why you add before dividing.

   3 marks

   Replace the heights: $m=\frac{12+15+18}{3}$. (1 mark)

   Add the numerator, then divide: $m=\frac{45}{3}=15$ cm. (1 mark)

   The fraction bar means divide the whole sum $a+b+c$ by 3, so that sum must be calculated first. (1 mark)

   The mean height is 15 cm. The fraction bar groups the entire sum in the numerator.
7. A rectangular label has length $l=9$ cm and width $w=4$ cm. Use $P=2l+2w$ to find its perimeter. State the unit and check your answer by adding its four side lengths.

   3 marks

   Substitute: $P=2\times 9+2\times 4$. (1 mark)

   Calculate: $P=18+8=26$ cm. Perimeter is a length. (1 mark)

   Add the four sides: $9+4+9+4=26$ cm, matching the formula. (1 mark)

   $P=26$ cm; $9+4+9+4=26$ cm confirms it.
8. A bike hire business uses $C=7h+4$, where $C$ is the total cost in dollars and $h$ is the number of hours. Find the cost for

   1. (a)2 hours and
   2. (b)4 hours. Mia says doubling 2 hours to 4 hours doubles the total cost. Use your results and the formula to explain whether she is correct.

   4 marks

   For $h=2$, $C=7\times 2+4=14+4=18$ dollars. (1 mark)

   For $h=4$, $C=7\times 4+4=28+4=32$ dollars. (1 mark)

   Twice the 2-hour cost is $2\times 18=36$ dollars, which is not 32 dollars. (1 mark)

   The hourly part doubles, but the constant 4-dollar charge is added only once for each hire. Doubling the whole original cost would count that fixed charge twice. (1 mark)

   1. (a)18 dollars;
   2. (b)32 dollars. Mia is incorrect: twice 18 dollars is 36 dollars, while the 4-dollar fixed charge is paid only once.