Practice · Average cost under a flat fee plus surcharge · Sensim Math

Variant 1 easy

A company is ordering promotional ballpoint pens. An order of $50$ pens or fewer costs a flat base fee of $6000$ won. For an order of more than $50$ pens, each pen beyond the first $50$ costs an extra $60$ won. For the average cost per pen to be less than $80$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $50$ pens cost a flat $6000$ won; each pen beyond $50$ adds $60$ won. Find the smallest order for which the average cost per pen is below $80$ won.

Givens

Orders of $50$ pens or fewer cost $6000$ won in total.
Each pen beyond the first $50$ costs an extra $60$ won.
The average cost per pen must be less than $80$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $50$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $50$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $50$ , write the total cost as an expression and set average $<80$ .

3 · Execute4 carry out the plan

1Rule out orders of 50 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

6000

divided by the number of pens, which is smallest when the order is

50

pens.

\frac{6000}{50} = 120 > 80

Even at its cheapest, the flat-fee range averages

120

won per pen, so the answer must be above

50

.

2Total cost above 50 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 50

, the extra pens number

x - 50

.

6000 + 60(x - 50) = 60x + 3000

Past

50

pens, the cost is a fixed part plus

60

won per pen.

3Average cost below 80

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{60x + 3000}{x} < 80 \;\Rightarrow\; 60x + 3000 < 80x \;\Rightarrow\; 3000 < 20x \;\Rightarrow\; x > 150

Each extra pen costs only

60

won, which pulls the average down toward

60

; it drops below

80

once there are more than

150

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

150

is

151

. Check it against

150

.

\frac{12060}{151} \approx 79.87 < 80, \qquad \frac{12000}{150} = 80

At

150

pens the average is exactly

80

, which is not less than

80

.

Answer:

151

pens

4 · Reviewdoes it hold up?

$151$ pens cost $6000 + 101 \times 60 = 12060$ won, and $12060 \div 151 \approx 79.87$ won, just under $80$ . The answer is above $50$ , as the first step required.

Another way: Compare with a plain $80$ won per pen. Above $50$ pens the cost is $60x + 3000$ , while $80$ per pen costs $80x$ . The pen order beats it once the $20$ won saved on every pen covers the fixed $3000$ : $20x > 3000$ , so $x > 150$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $60x + 3000 < 80x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 2 easy

A company is ordering promotional ballpoint pens. An order of $50$ pens or fewer costs a flat base fee of $10000$ won. For an order of more than $50$ pens, each pen beyond the first $50$ costs an extra $100$ won. For the average cost per pen to be less than $150$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $50$ pens cost a flat $10000$ won; each pen beyond $50$ adds $100$ won. Find the smallest order for which the average cost per pen is below $150$ won.

Givens

Orders of $50$ pens or fewer cost $10000$ won in total.
Each pen beyond the first $50$ costs an extra $100$ won.
The average cost per pen must be less than $150$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $50$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $50$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $50$ , write the total cost as an expression and set average $<150$ .

3 · Execute4 carry out the plan

1Rule out orders of 50 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

10000

divided by the number of pens, which is smallest when the order is

50

pens.

\frac{10000}{50} = 200 > 150

Even at its cheapest, the flat-fee range averages

200

won per pen, so the answer must be above

50

.

2Total cost above 50 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 50

, the extra pens number

x - 50

.

10000 + 100(x - 50) = 100x + 5000

Past

50

pens, the cost is a fixed part plus

100

won per pen.

3Average cost below 150

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{100x + 5000}{x} < 150 \;\Rightarrow\; 100x + 5000 < 150x \;\Rightarrow\; 5000 < 50x \;\Rightarrow\; x > 100

Each extra pen costs only

100

won, which pulls the average down toward

100

; it drops below

150

once there are more than

100

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

100

is

101

. Check it against

100

.

\frac{15100}{101} \approx 149.50 < 150, \qquad \frac{15000}{100} = 150

At

100

pens the average is exactly

150

, which is not less than

150

.

Answer:

101

pens

4 · Reviewdoes it hold up?

$101$ pens cost $10000 + 51 \times 100 = 15100$ won, and $15100 \div 101 \approx 149.50$ won, just under $150$ . The answer is above $50$ , as the first step required.

Another way: Compare with a plain $150$ won per pen. Above $50$ pens the cost is $100x + 5000$ , while $150$ per pen costs $150x$ . The pen order beats it once the $50$ won saved on every pen covers the fixed $5000$ : $50x > 5000$ , so $x > 100$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $100x + 5000 < 150x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 3 easy

A company is ordering promotional ballpoint pens. An order of $100$ pens or fewer costs a flat base fee of $12000$ won. For an order of more than $100$ pens, each pen beyond the first $100$ costs an extra $80$ won. For the average cost per pen to be less than $100$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $100$ pens cost a flat $12000$ won; each pen beyond $100$ adds $80$ won. Find the smallest order for which the average cost per pen is below $100$ won.

Givens

Orders of $100$ pens or fewer cost $12000$ won in total.
Each pen beyond the first $100$ costs an extra $80$ won.
The average cost per pen must be less than $100$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $100$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $100$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $100$ , write the total cost as an expression and set average $<100$ .

3 · Execute4 carry out the plan

1Rule out orders of 100 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

12000

divided by the number of pens, which is smallest when the order is

100

pens.

\frac{12000}{100} = 120 > 100

Even at its cheapest, the flat-fee range averages

120

won per pen, so the answer must be above

100

.

2Total cost above 100 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 100

, the extra pens number

x - 100

.

12000 + 80(x - 100) = 80x + 4000

Past

100

pens, the cost is a fixed part plus

80

won per pen.

3Average cost below 100

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{80x + 4000}{x} < 100 \;\Rightarrow\; 80x + 4000 < 100x \;\Rightarrow\; 4000 < 20x \;\Rightarrow\; x > 200

Each extra pen costs only

80

won, which pulls the average down toward

80

; it drops below

100

once there are more than

200

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

200

is

201

. Check it against

200

.

\frac{20080}{201} \approx 99.90 < 100, \qquad \frac{20000}{200} = 100

At

200

pens the average is exactly

100

, which is not less than

100

.

Answer:

201

pens

4 · Reviewdoes it hold up?

$201$ pens cost $12000 + 101 \times 80 = 20080$ won, and $20080 \div 201 \approx 99.90$ won, just under $100$ . The answer is above $100$ , as the first step required.

Another way: Compare with a plain $100$ won per pen. Above $100$ pens the cost is $80x + 4000$ , while $100$ per pen costs $100x$ . The pen order beats it once the $20$ won saved on every pen covers the fixed $4000$ : $20x > 4000$ , so $x > 200$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $80x + 4000 < 100x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 4 easy

A company is ordering promotional ballpoint pens. An order of $100$ pens or fewer costs a flat base fee of $15000$ won. For an order of more than $100$ pens, each pen beyond the first $100$ costs an extra $50$ won. For the average cost per pen to be less than $100$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $100$ pens cost a flat $15000$ won; each pen beyond $100$ adds $50$ won. Find the smallest order for which the average cost per pen is below $100$ won.

Givens

Orders of $100$ pens or fewer cost $15000$ won in total.
Each pen beyond the first $100$ costs an extra $50$ won.
The average cost per pen must be less than $100$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $100$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $100$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $100$ , write the total cost as an expression and set average $<100$ .

3 · Execute4 carry out the plan

1Rule out orders of 100 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

15000

divided by the number of pens, which is smallest when the order is

100

pens.

\frac{15000}{100} = 150 > 100

Even at its cheapest, the flat-fee range averages

150

won per pen, so the answer must be above

100

.

2Total cost above 100 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 100

, the extra pens number

x - 100

.

15000 + 50(x - 100) = 50x + 10000

Past

100

pens, the cost is a fixed part plus

50

won per pen.

3Average cost below 100

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{50x + 10000}{x} < 100 \;\Rightarrow\; 50x + 10000 < 100x \;\Rightarrow\; 10000 < 50x \;\Rightarrow\; x > 200

Each extra pen costs only

50

won, which pulls the average down toward

50

; it drops below

100

once there are more than

200

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

200

is

201

. Check it against

200

.

\frac{20050}{201} \approx 99.75 < 100, \qquad \frac{20000}{200} = 100

At

200

pens the average is exactly

100

, which is not less than

100

.

Answer:

201

pens

4 · Reviewdoes it hold up?

$201$ pens cost $15000 + 101 \times 50 = 20050$ won, and $20050 \div 201 \approx 99.75$ won, just under $100$ . The answer is above $100$ , as the first step required.

Another way: Compare with a plain $100$ won per pen. Above $100$ pens the cost is $50x + 10000$ , while $100$ per pen costs $100x$ . The pen order beats it once the $50$ won saved on every pen covers the fixed $10000$ : $50x > 10000$ , so $x > 200$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $50x + 10000 < 100x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 5 medium

A company is ordering promotional ballpoint pens. An order of $80$ pens or fewer costs a flat base fee of $16000$ won. For an order of more than $80$ pens, each pen beyond the first $80$ costs an extra $100$ won. For the average cost per pen to be less than $140$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $80$ pens cost a flat $16000$ won; each pen beyond $80$ adds $100$ won. Find the smallest order for which the average cost per pen is below $140$ won.

Givens

Orders of $80$ pens or fewer cost $16000$ won in total.
Each pen beyond the first $80$ costs an extra $100$ won.
The average cost per pen must be less than $140$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $80$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $80$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $80$ , write the total cost as an expression and set average $<140$ .

3 · Execute4 carry out the plan

1Rule out orders of 80 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

16000

divided by the number of pens, which is smallest when the order is

80

pens.

\frac{16000}{80} = 200 > 140

Even at its cheapest, the flat-fee range averages

200

won per pen, so the answer must be above

80

.

2Total cost above 80 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 80

, the extra pens number

x - 80

.

16000 + 100(x - 80) = 100x + 8000

Past

80

pens, the cost is a fixed part plus

100

won per pen.

3Average cost below 140

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{100x + 8000}{x} < 140 \;\Rightarrow\; 100x + 8000 < 140x \;\Rightarrow\; 8000 < 40x \;\Rightarrow\; x > 200

Each extra pen costs only

100

won, which pulls the average down toward

100

; it drops below

140

once there are more than

200

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

200

is

201

. Check it against

200

.

\frac{28100}{201} \approx 139.80 < 140, \qquad \frac{28000}{200} = 140

At

200

pens the average is exactly

140

, which is not less than

140

.

Answer:

201

pens

4 · Reviewdoes it hold up?

$201$ pens cost $16000 + 121 \times 100 = 28100$ won, and $28100 \div 201 \approx 139.80$ won, just under $140$ . The answer is above $80$ , as the first step required.

Another way: Compare with a plain $140$ won per pen. Above $80$ pens the cost is $100x + 8000$ , while $140$ per pen costs $140x$ . The pen order beats it once the $40$ won saved on every pen covers the fixed $8000$ : $40x > 8000$ , so $x > 200$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $100x + 8000 < 140x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 6 medium

A company is ordering promotional ballpoint pens. An order of $100$ pens or fewer costs a flat base fee of $18000$ won. For an order of more than $100$ pens, each pen beyond the first $100$ costs an extra $60$ won. For the average cost per pen to be less than $90$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $100$ pens cost a flat $18000$ won; each pen beyond $100$ adds $60$ won. Find the smallest order for which the average cost per pen is below $90$ won.

Givens

Orders of $100$ pens or fewer cost $18000$ won in total.
Each pen beyond the first $100$ costs an extra $60$ won.
The average cost per pen must be less than $90$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $100$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $100$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $100$ , write the total cost as an expression and set average $<90$ .

3 · Execute4 carry out the plan

1Rule out orders of 100 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

18000

divided by the number of pens, which is smallest when the order is

100

pens.

\frac{18000}{100} = 180 > 90

Even at its cheapest, the flat-fee range averages

180

won per pen, so the answer must be above

100

.

2Total cost above 100 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 100

, the extra pens number

x - 100

.

18000 + 60(x - 100) = 60x + 12000

Past

100

pens, the cost is a fixed part plus

60

won per pen.

3Average cost below 90

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{60x + 12000}{x} < 90 \;\Rightarrow\; 60x + 12000 < 90x \;\Rightarrow\; 12000 < 30x \;\Rightarrow\; x > 400

Each extra pen costs only

60

won, which pulls the average down toward

60

; it drops below

90

once there are more than

400

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

400

is

401

. Check it against

400

.

\frac{36060}{401} \approx 89.93 < 90, \qquad \frac{36000}{400} = 90

At

400

pens the average is exactly

90

, which is not less than

90

.

Answer:

401

pens

4 · Reviewdoes it hold up?

$401$ pens cost $18000 + 301 \times 60 = 36060$ won, and $36060 \div 401 \approx 89.93$ won, just under $90$ . The answer is above $100$ , as the first step required.

Another way: Compare with a plain $90$ won per pen. Above $100$ pens the cost is $60x + 12000$ , while $90$ per pen costs $90x$ . The pen order beats it once the $30$ won saved on every pen covers the fixed $12000$ : $30x > 12000$ , so $x > 400$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $60x + 12000 < 90x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 7 medium

A company is ordering promotional ballpoint pens. An order of $100$ pens or fewer costs a flat base fee of $20000$ won. For an order of more than $100$ pens, each pen beyond the first $100$ costs an extra $100$ won. For the average cost per pen to be less than $150$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $100$ pens cost a flat $20000$ won; each pen beyond $100$ adds $100$ won. Find the smallest order for which the average cost per pen is below $150$ won.

Givens

Orders of $100$ pens or fewer cost $20000$ won in total.
Each pen beyond the first $100$ costs an extra $100$ won.
The average cost per pen must be less than $150$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $100$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $100$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $100$ , write the total cost as an expression and set average $<150$ .

3 · Execute4 carry out the plan

1Rule out orders of 100 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

20000

divided by the number of pens, which is smallest when the order is

100

pens.

\frac{20000}{100} = 200 > 150

Even at its cheapest, the flat-fee range averages

200

won per pen, so the answer must be above

100

.

2Total cost above 100 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 100

, the extra pens number

x - 100

.

20000 + 100(x - 100) = 100x + 10000

Past

100

pens, the cost is a fixed part plus

100

won per pen.

3Average cost below 150

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{100x + 10000}{x} < 150 \;\Rightarrow\; 100x + 10000 < 150x \;\Rightarrow\; 10000 < 50x \;\Rightarrow\; x > 200

Each extra pen costs only

100

won, which pulls the average down toward

100

; it drops below

150

once there are more than

200

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

200

is

201

. Check it against

200

.

\frac{30100}{201} \approx 149.75 < 150, \qquad \frac{30000}{200} = 150

At

200

pens the average is exactly

150

, which is not less than

150

.

Answer:

201

pens

4 · Reviewdoes it hold up?

$201$ pens cost $20000 + 101 \times 100 = 30100$ won, and $30100 \div 201 \approx 149.75$ won, just under $150$ . The answer is above $100$ , as the first step required.

Another way: Compare with a plain $150$ won per pen. Above $100$ pens the cost is $100x + 10000$ , while $150$ per pen costs $150x$ . The pen order beats it once the $50$ won saved on every pen covers the fixed $10000$ : $50x > 10000$ , so $x > 200$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $100x + 10000 < 150x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 8 medium

A company is ordering promotional ballpoint pens. An order of $100$ pens or fewer costs a flat base fee of $25000$ won. For an order of more than $100$ pens, each pen beyond the first $100$ costs an extra $100$ won. For the average cost per pen to be less than $200$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $100$ pens cost a flat $25000$ won; each pen beyond $100$ adds $100$ won. Find the smallest order for which the average cost per pen is below $200$ won.

Givens

Orders of $100$ pens or fewer cost $25000$ won in total.
Each pen beyond the first $100$ costs an extra $100$ won.
The average cost per pen must be less than $200$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $100$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $100$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $100$ , write the total cost as an expression and set average $<200$ .

3 · Execute4 carry out the plan

1Rule out orders of 100 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

25000

divided by the number of pens, which is smallest when the order is

100

pens.

\frac{25000}{100} = 250 > 200

Even at its cheapest, the flat-fee range averages

250

won per pen, so the answer must be above

100

.

2Total cost above 100 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 100

, the extra pens number

x - 100

.

25000 + 100(x - 100) = 100x + 15000

Past

100

pens, the cost is a fixed part plus

100

won per pen.

3Average cost below 200

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{100x + 15000}{x} < 200 \;\Rightarrow\; 100x + 15000 < 200x \;\Rightarrow\; 15000 < 100x \;\Rightarrow\; x > 150

Each extra pen costs only

100

won, which pulls the average down toward

100

; it drops below

200

once there are more than

150

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

150

is

151

. Check it against

150

.

\frac{30100}{151} \approx 199.34 < 200, \qquad \frac{30000}{150} = 200

At

150

pens the average is exactly

200

, which is not less than

200

.

Answer:

151

pens

4 · Reviewdoes it hold up?

$151$ pens cost $25000 + 51 \times 100 = 30100$ won, and $30100 \div 151 \approx 199.34$ won, just under $200$ . The answer is above $100$ , as the first step required.

Another way: Compare with a plain $200$ won per pen. Above $100$ pens the cost is $100x + 15000$ , while $200$ per pen costs $200x$ . The pen order beats it once the $100$ won saved on every pen covers the fixed $15000$ : $100x > 15000$ , so $x > 150$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $100x + 15000 < 200x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 9 hard

A company is ordering promotional ballpoint pens. An order of $150$ pens or fewer costs a flat base fee of $30000$ won. For an order of more than $150$ pens, each pen beyond the first $150$ costs an extra $120$ won. For the average cost per pen to be less than $160$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $150$ pens cost a flat $30000$ won; each pen beyond $150$ adds $120$ won. Find the smallest order for which the average cost per pen is below $160$ won.

Givens

Orders of $150$ pens or fewer cost $30000$ won in total.
Each pen beyond the first $150$ costs an extra $120$ won.
The average cost per pen must be less than $160$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $150$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $150$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $150$ , write the total cost as an expression and set average $<160$ .

3 · Execute4 carry out the plan

1Rule out orders of 150 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

30000

divided by the number of pens, which is smallest when the order is

150

pens.

\frac{30000}{150} = 200 > 160

Even at its cheapest, the flat-fee range averages

200

won per pen, so the answer must be above

150

.

2Total cost above 150 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 150

, the extra pens number

x - 150

.

30000 + 120(x - 150) = 120x + 12000

Past

150

pens, the cost is a fixed part plus

120

won per pen.

3Average cost below 160

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{120x + 12000}{x} < 160 \;\Rightarrow\; 120x + 12000 < 160x \;\Rightarrow\; 12000 < 40x \;\Rightarrow\; x > 300

Each extra pen costs only

120

won, which pulls the average down toward

120

; it drops below

160

once there are more than

300

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

300

is

301

. Check it against

300

.

\frac{48120}{301} \approx 159.87 < 160, \qquad \frac{48000}{300} = 160

At

300

pens the average is exactly

160

, which is not less than

160

.

Answer:

301

pens

4 · Reviewdoes it hold up?

$301$ pens cost $30000 + 151 \times 120 = 48120$ won, and $48120 \div 301 \approx 159.87$ won, just under $160$ . The answer is above $150$ , as the first step required.

Another way: Compare with a plain $160$ won per pen. Above $150$ pens the cost is $120x + 12000$ , while $160$ per pen costs $160x$ . The pen order beats it once the $40$ won saved on every pen covers the fixed $12000$ : $40x > 12000$ , so $x > 300$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $120x + 12000 < 160x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 10 hard

A company is ordering promotional ballpoint pens. An order of $100$ pens or fewer costs a flat base fee of $30000$ won. For an order of more than $100$ pens, each pen beyond the first $100$ costs an extra $200$ won. For the average cost per pen to be less than $250$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $100$ pens cost a flat $30000$ won; each pen beyond $100$ adds $200$ won. Find the smallest order for which the average cost per pen is below $250$ won.

Givens

Orders of $100$ pens or fewer cost $30000$ won in total.
Each pen beyond the first $100$ costs an extra $200$ won.
The average cost per pen must be less than $250$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $100$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $100$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $100$ , write the total cost as an expression and set average $<250$ .

3 · Execute4 carry out the plan

1Rule out orders of 100 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

30000

divided by the number of pens, which is smallest when the order is

100

pens.

\frac{30000}{100} = 300 > 250

Even at its cheapest, the flat-fee range averages

300

won per pen, so the answer must be above

100

.

2Total cost above 100 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 100

, the extra pens number

x - 100

.

30000 + 200(x - 100) = 200x + 10000

Past

100

pens, the cost is a fixed part plus

200

won per pen.

3Average cost below 250

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{200x + 10000}{x} < 250 \;\Rightarrow\; 200x + 10000 < 250x \;\Rightarrow\; 10000 < 50x \;\Rightarrow\; x > 200

Each extra pen costs only

200

won, which pulls the average down toward

200

; it drops below

250

once there are more than

200

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

200

is

201

. Check it against

200

.

\frac{50200}{201} \approx 249.75 < 250, \qquad \frac{50000}{200} = 250

At

200

pens the average is exactly

250

, which is not less than

250

.

Answer:

201

pens

4 · Reviewdoes it hold up?

$201$ pens cost $30000 + 101 \times 200 = 50200$ won, and $50200 \div 201 \approx 249.75$ won, just under $250$ . The answer is above $100$ , as the first step required.

Another way: Compare with a plain $250$ won per pen. Above $100$ pens the cost is $200x + 10000$ , while $250$ per pen costs $250x$ . The pen order beats it once the $50$ won saved on every pen covers the fixed $10000$ : $50x > 10000$ , so $x > 200$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $200x + 10000 < 250x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 11 hard

A company is ordering promotional ballpoint pens. An order of $200$ pens or fewer costs a flat base fee of $30000$ won. For an order of more than $200$ pens, each pen beyond the first $200$ costs an extra $100$ won. For the average cost per pen to be less than $120$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $200$ pens cost a flat $30000$ won; each pen beyond $200$ adds $100$ won. Find the smallest order for which the average cost per pen is below $120$ won.

Givens

Orders of $200$ pens or fewer cost $30000$ won in total.
Each pen beyond the first $200$ costs an extra $100$ won.
The average cost per pen must be less than $120$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $200$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $200$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $200$ , write the total cost as an expression and set average $<120$ .

3 · Execute4 carry out the plan

1Rule out orders of 200 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

30000

divided by the number of pens, which is smallest when the order is

200

pens.

\frac{30000}{200} = 150 > 120

Even at its cheapest, the flat-fee range averages

150

won per pen, so the answer must be above

200

.

2Total cost above 200 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 200

, the extra pens number

x - 200

.

30000 + 100(x - 200) = 100x + 10000

Past

200

pens, the cost is a fixed part plus

100

won per pen.

3Average cost below 120

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{100x + 10000}{x} < 120 \;\Rightarrow\; 100x + 10000 < 120x \;\Rightarrow\; 10000 < 20x \;\Rightarrow\; x > 500

Each extra pen costs only

100

won, which pulls the average down toward

100

; it drops below

120

once there are more than

500

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

500

is

501

. Check it against

500

.

\frac{60100}{501} \approx 119.96 < 120, \qquad \frac{60000}{500} = 120

At

500

pens the average is exactly

120

, which is not less than

120

.

Answer:

501

pens

4 · Reviewdoes it hold up?

$501$ pens cost $30000 + 301 \times 100 = 60100$ won, and $60100 \div 501 \approx 119.96$ won, just under $120$ . The answer is above $200$ , as the first step required.

Another way: Compare with a plain $120$ won per pen. Above $200$ pens the cost is $100x + 10000$ , while $120$ per pen costs $120x$ . The pen order beats it once the $20$ won saved on every pen covers the fixed $10000$ : $20x > 10000$ , so $x > 500$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $100x + 10000 < 120x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Variant 12 hard

A company is ordering promotional ballpoint pens. An order of $200$ pens or fewer costs a flat base fee of $40000$ won. For an order of more than $200$ pens, each pen beyond the first $200$ costs an extra $150$ won. For the average cost per pen to be less than $175$ won, at least how many pens must be ordered?

Your answer

Show solution

1 · Understandwhat's really being asked

Up to $200$ pens cost a flat $40000$ won; each pen beyond $200$ adds $150$ won. Find the smallest order for which the average cost per pen is below $175$ won.

Givens

Orders of $200$ pens or fewer cost $40000$ won in total.
Each pen beyond the first $200$ costs an extra $150$ won.
The average cost per pen must be less than $175$ won.

Unknowns

The smallest number of pens to order.

Constraints

The number of pens is a whole number.
The pricing rule changes at $200$ pens, so the two ranges must be handled separately.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#3 Eliminate Possibilities#6 Guess and Check

Split the order sizes at $200$ . In the flat-fee range the average is far too high, so that range drops out. In the range above $200$ , write the total cost as an expression and set average $<175$ .

3 · Execute4 carry out the plan

1Rule out orders of 200 or fewer

#3 Eliminate Possibilities 7.EE.B.4

With the flat fee, the average is

40000

divided by the number of pens, which is smallest when the order is

200

pens.

\frac{40000}{200} = 200 > 175

Even at its cheapest, the flat-fee range averages

200

won per pen, so the answer must be above

200

.

2Total cost above 200 pens

#7 Identify Subproblems 7.EE.B.4

For

x

pens with

x > 200

, the extra pens number

x - 200

.

40000 + 150(x - 200) = 150x + 10000

Past

200

pens, the cost is a fixed part plus

150

won per pen.

3Average cost below 175

#13 Convert to Algebra 7.EE.B.4

Average cost is total cost divided by

x

. Since

x > 0

, multiply both sides by

x

.

\frac{150x + 10000}{x} < 175 \;\Rightarrow\; 150x + 10000 < 175x \;\Rightarrow\; 10000 < 25x \;\Rightarrow\; x > 400

Each extra pen costs only

150

won, which pulls the average down toward

150

; it drops below

175

once there are more than

400

pens.

4Smallest whole number and check

#6 Guess and Check 7.EE.B.4

The smallest whole number greater than

400

is

401

. Check it against

400

.

\frac{70150}{401} \approx 174.94 < 175, \qquad \frac{70000}{400} = 175

At

400

pens the average is exactly

175

, which is not less than

175

.

Answer:

401

pens

4 · Reviewdoes it hold up?

$401$ pens cost $40000 + 201 \times 150 = 70150$ won, and $70150 \div 401 \approx 174.94$ won, just under $175$ . The answer is above $200$ , as the first step required.

Another way: Compare with a plain $175$ won per pen. Above $200$ pens the cost is $150x + 10000$ , while $175$ per pen costs $175x$ . The pen order beats it once the $25$ won saved on every pen covers the fixed $10000$ : $25x > 10000$ , so $x > 400$ .

Standardsmin grade 7

7.EE.B.4 Use variables to represent quantities and construct simple equations and inequalities — Writing the total cost and the average-cost condition with a variable, ruling out the flat-fee range, and solving $150x + 10000 < 175x$.

💡Takeaway. When a price rule changes at a cutoff, check each side of the cutoff separately -- one side may be ruled out right away.

Average cost under a flat fee plus surcharge · 12 practice problems

Generated variants — 12

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Where next?