Practice · The biggest saving and the biggest discount rate need not be the same item · Sensim Math

Variant 1 easy answer: The eraser

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Eraser	Notebook	Ruler
Original price (won)	$800$	$2500$	$1000$
Sale price (won)	$600$	$2000$	$850$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The eraser was 800 won and is now 600 won.
The notebook was 2500 won and is now 2000 won.
The ruler was 1000 won and is now 850 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

800 - 600 = 200,\quad 2500 - 2000 = 500,\quad 1000 - 850 = 150

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

200 \div 800,\quad 500 \div 2500,\quad 150 \div 1000

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

25\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The eraser's rate is the highest of the three.

25\%

The eraser is discounted most.

Answer: The eraser

4 · Reviewdoes it hold up?

Checking against the raw savings: the notebook was cut by the most won, 500, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the eraser would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 2 easy answer: The colored pencil

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Colored pencil	Set square	Memo pad
Original price (won)	$1500$	$4000$	$1200$
Sale price (won)	$1050$	$3000$	$960$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The colored pencil was 1500 won and is now 1050 won.
The set square was 4000 won and is now 3000 won.
The memo pad was 1200 won and is now 960 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

1500 - 1050 = 450,\quad 4000 - 3000 = 1000,\quad 1200 - 960 = 240

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

450 \div 1500,\quad 1000 \div 4000,\quad 240 \div 1200

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

30\%,\quad 25\%,\quad 20\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The colored pencil's rate is the highest of the three.

30\%

The colored pencil is discounted most.

Answer: The colored pencil

4 · Reviewdoes it hold up?

Checking against the raw savings: the set square was cut by the most won, 1000, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the colored pencil would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 3 easy answer: The notepad

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Notepad	Folder	Rubber band
Original price (won)	$1800$	$4500$	$300$
Sale price (won)	$1350$	$3600$	$255$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The notepad was 1800 won and is now 1350 won.
The folder was 4500 won and is now 3600 won.
The rubber band was 300 won and is now 255 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

1800 - 1350 = 450,\quad 4500 - 3600 = 900,\quad 300 - 255 = 45

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

450 \div 1800,\quad 900 \div 4500,\quad 45 \div 300

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

25\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The notepad's rate is the highest of the three.

25\%

The notepad is discounted most.

Answer: The notepad

4 · Reviewdoes it hold up?

Checking against the raw savings: the folder was cut by the most won, 900, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the notepad would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 4 easy answer: The glue stick

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Glue stick	Sketchbook	Sticker
Original price (won)	$1200$	$5000$	$600$
Sale price (won)	$900$	$4000$	$510$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The glue stick was 1200 won and is now 900 won.
The sketchbook was 5000 won and is now 4000 won.
The sticker was 600 won and is now 510 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

1200 - 900 = 300,\quad 5000 - 4000 = 1000,\quad 600 - 510 = 90

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

300 \div 1200,\quad 1000 \div 5000,\quad 90 \div 600

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

25\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The glue stick's rate is the highest of the three.

25\%

The glue stick is discounted most.

Answer: The glue stick

4 · Reviewdoes it hold up?

Checking against the raw savings: the sketchbook was cut by the most won, 1000, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the glue stick would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 5 medium answer: The pen

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Pen	File box	Tape
Original price (won)	$2000$	$6000$	$1500$
Sale price (won)	$1400$	$4800$	$1275$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The pen was 2000 won and is now 1400 won.
The file box was 6000 won and is now 4800 won.
The tape was 1500 won and is now 1275 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

2000 - 1400 = 600,\quad 6000 - 4800 = 1200,\quad 1500 - 1275 = 225

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

600 \div 2000,\quad 1200 \div 6000,\quad 225 \div 1500

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

30\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The pen's rate is the highest of the three.

30\%

The pen is discounted most.

Answer: The pen

4 · Reviewdoes it hold up?

Checking against the raw savings: the file box was cut by the most won, 1200, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the pen would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 6 medium answer: The highlighter

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Highlighter	Desk mat	Label
Original price (won)	$1000$	$7000$	$400$
Sale price (won)	$750$	$5600$	$340$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The highlighter was 1000 won and is now 750 won.
The desk mat was 7000 won and is now 5600 won.
The label was 400 won and is now 340 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

1000 - 750 = 250,\quad 7000 - 5600 = 1400,\quad 400 - 340 = 60

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

250 \div 1000,\quad 1400 \div 7000,\quad 60 \div 400

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

25\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The highlighter's rate is the highest of the three.

25\%

The highlighter is discounted most.

Answer: The highlighter

4 · Reviewdoes it hold up?

Checking against the raw savings: the desk mat was cut by the most won, 1400, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the highlighter would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 7 medium answer: The crayon

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Crayon	Paint set	Pencil case
Original price (won)	$3000$	$8000$	$4000$
Sale price (won)	$2100$	$6400$	$3400$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The crayon was 3000 won and is now 2100 won.
The paint set was 8000 won and is now 6400 won.
The pencil case was 4000 won and is now 3400 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

3000 - 2100 = 900,\quad 8000 - 6400 = 1600,\quad 4000 - 3400 = 600

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

900 \div 3000,\quad 1600 \div 8000,\quad 600 \div 4000

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

30\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The crayon's rate is the highest of the three.

30\%

The crayon is discounted most.

Answer: The crayon

4 · Reviewdoes it hold up?

Checking against the raw savings: the paint set was cut by the most won, 1600, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the crayon would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 8 medium answer: The stapler

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Stapler	Scissors	Refill
Original price (won)	$5000$	$9000$	$700$
Sale price (won)	$3500$	$7200$	$595$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The stapler was 5000 won and is now 3500 won.
The scissors was 9000 won and is now 7200 won.
The refill was 700 won and is now 595 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

5000 - 3500 = 1500,\quad 9000 - 7200 = 1800,\quad 700 - 595 = 105

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

1500 \div 5000,\quad 1800 \div 9000,\quad 105 \div 700

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

30\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The stapler's rate is the highest of the three.

30\%

The stapler is discounted most.

Answer: The stapler

4 · Reviewdoes it hold up?

Checking against the raw savings: the scissors was cut by the most won, 1800, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the stapler would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 9 hard answer: The ink

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Ink	Printer paper	Envelope
Original price (won)	$2200$	$10000$	$600$
Sale price (won)	$1650$	$8000$	$510$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The ink was 2200 won and is now 1650 won.
The printer paper was 10000 won and is now 8000 won.
The envelope was 600 won and is now 510 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

2200 - 1650 = 550,\quad 10000 - 8000 = 2000,\quad 600 - 510 = 90

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

550 \div 2200,\quad 2000 \div 10000,\quad 90 \div 600

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

25\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The ink's rate is the highest of the three.

25\%

The ink is discounted most.

Answer: The ink

4 · Reviewdoes it hold up?

Checking against the raw savings: the printer paper was cut by the most won, 2000, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the ink would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 10 hard answer: The compass

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Compass	Calculator	Bookmark
Original price (won)	$1600$	$12000$	$900$
Sale price (won)	$1200$	$9600$	$765$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The compass was 1600 won and is now 1200 won.
The calculator was 12000 won and is now 9600 won.
The bookmark was 900 won and is now 765 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

1600 - 1200 = 400,\quad 12000 - 9600 = 2400,\quad 900 - 765 = 135

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

400 \div 1600,\quad 2400 \div 12000,\quad 135 \div 900

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

25\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The compass's rate is the highest of the three.

25\%

The compass is discounted most.

Answer: The compass

4 · Reviewdoes it hold up?

Checking against the raw savings: the calculator was cut by the most won, 2400, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the compass would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 11 hard answer: The chalk

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Chalk	Whiteboard	Magnet
Original price (won)	$1400$	$20000$	$800$
Sale price (won)	$980$	$16000$	$680$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The chalk was 1400 won and is now 980 won.
The whiteboard was 20000 won and is now 16000 won.
The magnet was 800 won and is now 680 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

1400 - 980 = 420,\quad 20000 - 16000 = 4000,\quad 800 - 680 = 120

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

420 \div 1400,\quad 4000 \div 20000,\quad 120 \div 800

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

30\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The chalk's rate is the highest of the three.

30\%

The chalk is discounted most.

Answer: The chalk

4 · Reviewdoes it hold up?

Checking against the raw savings: the whiteboard was cut by the most won, 4000, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the chalk would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

Variant 12 hard answer: The marker

The table shows the original price and the discounted price of items at a stationery shop. Which item has the highest discount rate?

Item	Marker	Backpack	Clip
Original price (won)	$2400$	$30000$	$500$
Sale price (won)	$1800$	$24000$	$425$

Show solution

1 · Understandwhat's really being asked

Three items each have an original price and a sale price. We want the one whose discount is the largest share of its own price.

Givens

The marker was 2400 won and is now 1800 won.
The backpack was 30000 won and is now 24000 won.
The clip was 500 won and is now 425 won.

Unknowns

Which item has the highest discount rate.

Constraints

A discount rate compares the cut with the original price, not with the other items.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #2 Make a Systematic List#15 Organize Information in More Ways

The prices are not comparable as they stand -- a 500 won cut means something different on a 1000 won item and a 5000 won one. Turn each cut into a share of its own original price, and then one column of numbers can be compared.

3 · Execute4 carry out the plan

1Find each saving

#2 Make a Systematic List 6.RP.A.3

Original price minus sale price.

2400 - 1800 = 600,\quad 30000 - 24000 = 6000,\quad 500 - 425 = 75

Three savings, in won.

2Divide each saving by its own price

#8 Analyze the Units 7.RP.A.3

That is what makes the three comparable: each becomes a share of the price it came off.

600 \div 2400,\quad 6000 \div 30000,\quad 75 \div 500

Now they are on the same footing.

3Write them as percents

#15 Organize Information in More Ways 7.RP.A.3

Multiplying each rate by 100 makes them easy to read.

25\%,\quad 20\%,\quad 15\%

One column of comparable numbers.

4Pick the largest

#2 Make a Systematic List 6.RP.A.3

The marker's rate is the highest of the three.

25\%

The marker is discounted most.

Answer: The marker

4 · Reviewdoes it hold up?

Checking against the raw savings: the backpack was cut by the most won, 6000, but it started from a higher price -- so the largest cut and the largest rate really are different items here.

Another way: Comparing sale price to original price instead gives the share kept rather than the share saved; the marker would then be the lowest, which is the same conclusion upside down.

Standardsmin grade 7

7.RP.A.3 Use proportional relationships to solve multi-step ratio and percent problems — Turning each discount into a percentage of its own base.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Comparing three rates once they share a base.

💡Takeaway. How much you save and how good the deal is are different questions. Compare each saving with its own price.

The biggest saving and the biggest discount rate need not be the same item · 12 practice problems

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