Algebra, NAPX-E4-CA30 SA

Question

{{{question}}}

Worked Solution

{{{workedSolution}}}

Variant 0

DifficultyLevel

706

Question

A teacher is choosing two students from a group of 3 to ring the school bell.

There are a total of 3 different combinations that are possible.

The formula below gives the total number of combinations CC if the teacher is choosing from SS students.

C=0.5S(S−1)C=0.5S (S − 1)


What is the value of SS if the total possible combinations CC is 153?

Worked Solution

Strategy 1

By trial and error:

If  S=12, C=0.5×12×11=66\ S=12,\ C=0.5×12×11=66

If  S=16, C=0.5×16×15=120\ S=16,\ C=0.5×16×15=120

If  S=18, C=0.5×18×17=153\ S=18,\ C=0.5×18×17=153 ✓\checkmark


Strategy 2 (advanced)

C=0.5S(S − 1)C=0.5S(S\ −\ 1)

153=0.5S2 − 0.5S153=0.5S^{2}\ −\ 0.5S

S2 − S − 306=0S^{2}\ −\ S\ −\ 306=0

(S − 18)(S+17)=0(S\ −\ 18)(S+17)=0

∴ S\therefore \ S = 18 ,  S>0\ S>0

Question Type

Answer Box

Variables

Variable nameVariable value
question
A teacher is choosing two students from a group of 3 to ring the school bell. There are a total of 3 different combinations that are possible. The formula below gives the total number of combinations $C$ if the teacher is choosing from $S$ students. >> $C=0.5S (S − 1)$
What is the value of $S$ if the total possible combinations $C$ is 153?
workedSolution
Strategy 1 By trial and error: If $\ S=12,\ C=0.5×12×11=66$ If $\ S=16,\ C=0.5×16×15=120$ If $\ S=18,\ C=0.5×18×17=153$ $\checkmark$
Strategy 2 (advanced) $C=0.5S(S\ −\ 1)$ $153=0.5S^{2}\ −\ 0.5S$ $S^{2}\ −\ S\ −\ 306=0$ $(S\ −\ 18)(S+17)=0$ $\therefore \ S$ = {{{correctAnswer0}}} , $\ S>0$
correctAnswer0
18
prefix0
suffix0

Answers

Specify one or more 'ANSWER' block(s) as exampled below.
Note: correctAnswer is required, the rest are optional. ("correctAnswer" is what the student would need to type in to the box to get the answer correct.)

For example:
correctAnswer: 123.40

And optionally, specify the following, but only if you need something different to the defaults: 'width' defaults to 5 if not present, and valid values are 3 to 10; 'prefix' and 'suffix' default to nothing if not present.
prefix: $
suffix: mm$^2$
width: 5

correctAnswerNcorrectAnswerValueAnswer
correctAnswer018

Variant 1

DifficultyLevel

708

Question

A coach is choosing two players from a group of 3 to play in the goals in the upcoming netball semi-final.

There are a total of 3 different combinations that are possible.

The formula below gives the total number of combinations CC if the coach is choosing from PP players.

C=0.5P(P−1)C=0.5P (P − 1)


What is the value of PP if the total possible combinations CC is 91?

Worked Solution

Strategy 1

By trial and error:

If  P=10, C=0.5×10×9=45\ P=10,\ C=0.5×10×9=45

If  P=12, C=0.5×12×11=66\ P=12,\ C=0.5×12×11=66

If  P=14, C=0.5×14×13=91\ P=14,\ C=0.5×14×13=91 ✓\checkmark


Strategy 2 (advanced)

C=0.5P(P − 1)C=0.5P(P\ −\ 1)

91=0.5P2 − 0.5P91=0.5P^{2}\ −\ 0.5P

P2 − P − 182=0P^{2}\ −\ P\ −\ 182=0

(P − 14)(P+13)=0(P\ −\ 14)(P+13)=0

∴ P\therefore \ P = 14 ,  P>0\ P>0

Question Type

Answer Box

Variables

Variable nameVariable value
question
A coach is choosing two players from a group of 3 to play in the goals in the upcoming netball semi-final. There are a total of 3 different combinations that are possible. The formula below gives the total number of combinations $C$ if the coach is choosing from $P$ players. >> $C=0.5P (P − 1)$
What is the value of $P$ if the total possible combinations $C$ is 91?
workedSolution
Strategy 1 By trial and error: If $\ P=10,\ C=0.5×10×9=45$ If $\ P=12,\ C=0.5×12×11=66$ If $\ P=14,\ C=0.5×14×13=91$ $\checkmark$
Strategy 2 (advanced) $C=0.5P(P\ −\ 1)$ $91=0.5P^{2}\ −\ 0.5P$ $P^{2}\ −\ P\ −\ 182=0$ $(P\ −\ 14)(P+13)=0$ $\therefore \ P$ = {{{correctAnswer0}}} , $\ P>0$
correctAnswer0
14
prefix0
suffix0

Answers

Specify one or more 'ANSWER' block(s) as exampled below.
Note: correctAnswer is required, the rest are optional. ("correctAnswer" is what the student would need to type in to the box to get the answer correct.)

For example:
correctAnswer: 123.40

And optionally, specify the following, but only if you need something different to the defaults: 'width' defaults to 5 if not present, and valid values are 3 to 10; 'prefix' and 'suffix' default to nothing if not present.
prefix: $
suffix: mm$^2$
width: 5

correctAnswerNcorrectAnswerValueAnswer
correctAnswer014

Variant 2

DifficultyLevel

708

Question

A teacher is choosing two students from a group of 3 to be class captains.

There are a total of 3 different combinations that are possible.

The formula below gives the total number of combinations CC if the teacher is choosing from SS students.

C=0.5S(S−1)C=0.5S (S − 1)


What is the value of SS if the total possible combinations CC is 406?

Worked Solution

Strategy 1

By trial and error:

If  S=25, C=0.5×25×24=300\ S=25,\ C=0.5×25×24=300

If  S=27, C=0.5×27×26=351\ S=27,\ C=0.5×27×26=351

If  S=29, C=0.5×29×28=406\ S=29,\ C=0.5×29×28=406 ✓\checkmark


Strategy 2 (advanced)

C=0.5S(S − 1)C=0.5S(S\ −\ 1)

406=0.5S2 − 0.5S406=0.5S^{2}\ −\ 0.5S

S2 − S − 812=0S^{2}\ −\ S\ −\ 812=0

(S − 29)(S+28)=0(S\ −\ 29)(S+28)=0

∴ S\therefore \ S = 29 ,  S>0\ S>0

Question Type

Answer Box

Variables

Variable nameVariable value
question
A teacher is choosing two students from a group of 3 to be class captains. There are a total of 3 different combinations that are possible. The formula below gives the total number of combinations $C$ if the teacher is choosing from $S$ students. >> $C=0.5S (S − 1)$
What is the value of $S$ if the total possible combinations $C$ is 406?
workedSolution
Strategy 1 By trial and error: If $\ S=25,\ C=0.5×25×24=300$ If $\ S=27,\ C=0.5×27×26=351$ If $\ S=29,\ C=0.5×29×28=406$ $\checkmark$
Strategy 2 (advanced) $C=0.5S(S\ −\ 1)$ $406=0.5S^{2}\ −\ 0.5S$ $S^{2}\ −\ S\ −\ 812=0$ $(S\ −\ 29)(S+28)=0$ $\therefore \ S$ = {{{correctAnswer0}}} , $\ S>0$
correctAnswer0
29
prefix0
suffix0

Answers

Specify one or more 'ANSWER' block(s) as exampled below.
Note: correctAnswer is required, the rest are optional. ("correctAnswer" is what the student would need to type in to the box to get the answer correct.)

For example:
correctAnswer: 123.40

And optionally, specify the following, but only if you need something different to the defaults: 'width' defaults to 5 if not present, and valid values are 3 to 10; 'prefix' and 'suffix' default to nothing if not present.
prefix: $
suffix: mm$^2$
width: 5

correctAnswerNcorrectAnswerValueAnswer
correctAnswer029

Variant 3

DifficultyLevel

704

Question

A gymnastics coach is choosing two gymnasts from a group of 3 to compete in the parallel bars event.

There are a total of 3 different combinations that are possible.

The formula below gives the total number of combinations CC if the coach is choosing from GG gymnasts.

C=0.5G(G−1)C=0.5G (G − 1)


What is the value of GG if the total possible combinations CC is 15?

Worked Solution

Strategy 1

By trial and error:

If  G=4, C=0.5×4×3=6\ G=4,\ C=0.5×4×3=6

If  G=5, C=0.5×5×4=10\ G=5,\ C=0.5×5×4=10

If  G=6, C=0.5×6×5=15\ G=6,\ C=0.5×6×5=15 ✓\checkmark


Strategy 2 (advanced)

C=0.5G(G − 1)C=0.5G(G\ −\ 1)

15=0.5G2 − 0.5G15=0.5G^{2}\ −\ 0.5G

G2 − G − 30=0G^{2}\ −\ G\ −\ 30=0

(G − 6)(G+5)=0(G\ −\ 6)(G+5)=0

∴ G\therefore \ G = 6 ,  G>0\ G>0

Question Type

Answer Box

Variables

Variable nameVariable value
question
A gymnastics coach is choosing two gymnasts from a group of 3 to compete in the parallel bars event. There are a total of 3 different combinations that are possible. The formula below gives the total number of combinations $C$ if the coach is choosing from $G$ gymnasts. >> $C=0.5G (G − 1)$
What is the value of $G$ if the total possible combinations $C$ is 15?
workedSolution
Strategy 1 By trial and error: If $\ G=4,\ C=0.5×4×3=6$ If $\ G=5,\ C=0.5×5×4=10$ If $\ G=6,\ C=0.5×6×5=15$ $\checkmark$
Strategy 2 (advanced) $C=0.5G(G\ −\ 1)$ $15=0.5G^{2}\ −\ 0.5G$ $G^{2}\ −\ G\ −\ 30=0$ $(G\ −\ 6)(G+5)=0$ $\therefore \ G$ = {{{correctAnswer0}}} , $\ G>0$
correctAnswer0
6
prefix0
suffix0

Answers

Specify one or more 'ANSWER' block(s) as exampled below.
Note: correctAnswer is required, the rest are optional. ("correctAnswer" is what the student would need to type in to the box to get the answer correct.)

For example:
correctAnswer: 123.40

And optionally, specify the following, but only if you need something different to the defaults: 'width' defaults to 5 if not present, and valid values are 3 to 10; 'prefix' and 'suffix' default to nothing if not present.
prefix: $
suffix: mm$^2$
width: 5

correctAnswerNcorrectAnswerValueAnswer
correctAnswer06

Variant 4

DifficultyLevel

706

Question

A teacher is choosing two students from a group of 3 to compete in the javelin event at the zone athletics carnival.

There are a total of 3 different combinations that are possible.

The formula below gives the total number of combinations CC if the teacher is choosing from SS students.

C=0.5S(S−1)C=0.5S (S − 1)


What is the value of SS if the total possible combinations CC is 78?

Worked Solution

Strategy 1

By trial and error:

If  S=9, C=0.5×9×8=36\ S=9,\ C=0.5×9×8=36

If  S=11, C=0.5×11×10=55\ S=11,\ C=0.5×11×10=55

If  S=13, C=0.5×13×12=78\ S=13,\ C=0.5×13×12=78 ✓\checkmark


Strategy 2 (advanced)

C=0.5S(S − 1)C=0.5S(S\ −\ 1)

78=0.5S2 − 0.5S78=0.5S^{2}\ −\ 0.5S

S2 − S − 156=0S^{2}\ −\ S\ −\ 156=0

(S − 13)(S+12)=0(S\ −\ 13)(S+12)=0

∴ S\therefore \ S = 13 ,  S>0\ S>0

Question Type

Answer Box

Variables

Variable nameVariable value
question
A teacher is choosing two students from a group of 3 to compete in the javelin event at the zone athletics carnival. There are a total of 3 different combinations that are possible. The formula below gives the total number of combinations $C$ if the teacher is choosing from $S$ students. >> $C=0.5S (S − 1)$
What is the value of $S$ if the total possible combinations $C$ is 78?
workedSolution
Strategy 1 By trial and error: If $\ S=9,\ C=0.5×9×8=36$ If $\ S=11,\ C=0.5×11×10=55$ If $\ S=13,\ C=0.5×13×12=78$ $\checkmark$
Strategy 2 (advanced) $C=0.5S(S\ −\ 1)$ $78=0.5S^{2}\ −\ 0.5S$ $S^{2}\ −\ S\ −\ 156=0$ $(S\ −\ 13)(S+12)=0$ $\therefore \ S$ = {{{correctAnswer0}}} , $\ S>0$
correctAnswer0
13
prefix0
suffix0

Answers

Specify one or more 'ANSWER' block(s) as exampled below.
Note: correctAnswer is required, the rest are optional. ("correctAnswer" is what the student would need to type in to the box to get the answer correct.)

For example:
correctAnswer: 123.40

And optionally, specify the following, but only if you need something different to the defaults: 'width' defaults to 5 if not present, and valid values are 3 to 10; 'prefix' and 'suffix' default to nothing if not present.
prefix: $
suffix: mm$^2$
width: 5

correctAnswerNcorrectAnswerValueAnswer
correctAnswer013

Variant 5

DifficultyLevel

707

Question

A teacher is choosing two students from a group of 3 to compete in the regional debating competition.

There are a total of 3 different combinations that are possible.

The formula below gives the total number of combinations CC if the teacher is choosing from SS students.

C=0.5S(S−1)C=0.5S (S − 1)


What is the value of SS if the total possible combinations CC is 300?

Worked Solution

Strategy 1

By trial and error:

If  S=21, C=0.5×21×20=210\ S=21,\ C=0.5×21×20=210

If  S=23, C=0.5×23×22=253\ S=23,\ C=0.5×23×22=253

If  S=25, C=0.5×25×24=300\ S=25,\ C=0.5×25×24=300 ✓\checkmark


Strategy 2 (advanced)

C=0.5S(S − 1)C=0.5S(S\ −\ 1)

300=0.5S2 − 0.5S300=0.5S^{2}\ −\ 0.5S

S2 − S − 600=0S^{2}\ −\ S\ −\ 600=0

(S − 25)(S+24)=0(S\ −\ 25)(S+24)=0

∴ S\therefore \ S = 25 ,  S>0\ S>0

Question Type

Answer Box

Variables

Variable nameVariable value
question
A teacher is choosing two students from a group of 3 to compete in the regional debating competition. There are a total of 3 different combinations that are possible. The formula below gives the total number of combinations $C$ if the teacher is choosing from $S$ students. >> $C=0.5S (S − 1)$
What is the value of $S$ if the total possible combinations $C$ is 300?
workedSolution
Strategy 1 By trial and error: If $\ S=21,\ C=0.5×21×20=210$ If $\ S=23,\ C=0.5×23×22=253$ If $\ S=25,\ C=0.5×25×24=300$ $\checkmark$
Strategy 2 (advanced) $C=0.5S(S\ −\ 1)$ $300=0.5S^{2}\ −\ 0.5S$ $S^{2}\ −\ S\ −\ 600=0$ $(S\ −\ 25)(S+24)=0$ $\therefore \ S$ = {{{correctAnswer0}}} , $\ S>0$
correctAnswer0
25
prefix0
suffix0

Answers

Specify one or more 'ANSWER' block(s) as exampled below.
Note: correctAnswer is required, the rest are optional. ("correctAnswer" is what the student would need to type in to the box to get the answer correct.)

For example:
correctAnswer: 123.40

And optionally, specify the following, but only if you need something different to the defaults: 'width' defaults to 5 if not present, and valid values are 3 to 10; 'prefix' and 'suffix' default to nothing if not present.
prefix: $
suffix: mm$^2$
width: 5

correctAnswerNcorrectAnswerValueAnswer
correctAnswer025

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