PAPER · REASONING · PRACTICE

Equations in context: quantities, rates and conditions

Choose representative cost, travel, work-rate, percentage, ratio and tiered-price models; verify the solution in context.

40 minutes · Year 7, upper volume · text pp. 67–78 · foundation lessons 34–39

Textbook reference: Year 7, upper volume · Chapter 3 · Linear equations

Chapter core practice: use after the referenced textbook lessons, or for chapter review. See the upper-volume plan

One worked example, one guided attempt, eight core questions, four follow-ups and two extensions. Write on paper; the teacher controls the reveal.

Teacher reference Arrange in My Classroom

Lesson routePhase

4′Check prerequisites
5′Observe and predict
8′Worked example and guided attempt
15′Independent practice
5′Discuss and correct
3′Exit questions

Observe · explain

A question before practising

For a bill with the first 20 units at 2 dollars and extra units at 3 dollars, compare a bill of 30 dollars with 49 dollars. Decide the applicable branch before solving.

Organise the key idea

An equation represents a stated equality between quantities. Name the unknown, keep units consistent and test whether the solution satisfies the original situation and any price branch.

Open the original chapter Open the interactive inquiry

Use the lesson question here for paper discussion; the original activity opens separately.

Worked example1

Two groups collected 32 cards in total. The larger group collected 8 more than the smaller. Let x be the smaller group’s number. Find both numbers.

Use the smaller quantity as x.
Express the larger quantity as x + 8.
Add both quantities, not just their difference.

The larger number is x + 8, so x + (x + 8) = 32.
2x + 8 = 32 gives x = 12; the other number is 20.
Check: 12 + 20 = 32 and 20 − 12 = 8.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Guided attempt2

A walker moves at 60 m/min. A runner starts from the same point 2 minutes later at 80 m/min in the same direction. Speeds stay constant. Let x be the runner’s time until catching up. Find x.

Walker: time ____ min; distance ____ m.
Runner: time ____ min; distance ____ m.
Equal-distance equation: ____.

The two travelling times are not equal.
The walker has travelled 2 minutes longer.
Use speed × time on both sides.

The walker’s time is x + 2, whereas the runner’s time is x.
Equal distances give 60(x + 2) = 80x.
60x + 120 = 80x gives x = 6; check 60 × 8 = 80 × 6 = 480.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Prerequisite3

One worker completes 1/6 of a job each hour and another completes 1/3. Find their combined hourly rate.

\frac{1}{6}+\frac{1}{3}

Add work per hour, not the separate completion times.
Use denominator 6.
Add 1/6 and 2/6.

The rates use the same whole job and the same hour unit.
1/6 + 1/3 = 1/6 + 2/6 = 1/2.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Prerequisite4

Profit is 20% of a 50-dollar cost. Find the profit, not the selling price.

\frac{20}{100}\times50

Identify the base before multiplying.
The base is 50, not the selling price.
Calculate one fifth of 50.

20% means 20/100 of the cost.
Multiply by 50 to get 10.

Check your work

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Independent practice5

Notebooks cost 4 dollars each, plus a single 3-dollar packing fee. The total is 27 dollars. Let x be the number of notebooks. Find x.

The packing fee is charged only once.
Subtract 3 from the total before dividing.
The count must be a nonnegative whole number.

4x + 3 = 27.
4x = 24 gives x = 6. Check 4 × 6 + 3 = 27.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Independent practice6

Two workers would finish the same job alone in 4 hours and 6 hours. They work together at constant independent rates. Let x be their time together. Find x.

Treat the whole job as 1.
Do not add 4 and 6 as work rates.
Use x/4 + x/6 = 1.

Each contribution is rate × time: x/4 and x/6; their sum is one job.
Multiply x/4 + x/6 = 1 by 12: 3x + 2x = 12, so x = 12/5.
Check: 3/5 + 2/5 = 1, and the time is shorter than either solo time.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Independent practice7

A product sells for 60 dollars at a profit of 20% of its cost. Let the cost be x dollars. Find x.

Profit percentage uses cost as its base.
Selling price is 120% of cost.
Divide 60 by 1.2.

Selling price = cost + 20% of cost = 1.2x.
1.2x = 60 gives x = 50. Check 50 + 10 = 60.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Independent practice8

The ratio of boys to girls is 2:3 and there are 35 pupils. Let x be the number of boys. Find x and check both counts.

x represents 2 ratio parts, not 1 part.
The girls are 3/2 times the boys.
Check that both final counts are whole numbers.

Girls number (3/2)x, so x + (3/2)x = 35.
(5/2)x = 35 gives x = 14; girls = 21.
Check 14 + 21 = 35 and 14:21 = 2:3.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Exit question9

An item marked at 80 dollars is sold at 25% off. That selling price is 20% above cost. Let cost be x dollars. Find x and identify both percentage bases.

25% off leaves 75% of the marked price.
20% above cost means 120% of cost.
Do not take 20% away from the selling price to find cost.

Selling price = 80 × 75% = 60 dollars.
60 = 1.2x, so x = 50.
Check 50 × 120% = 60 and 80 × 75% = 60.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Exit question10

The first 20 units cost 2 dollars each; additional units cost 3 dollars each. A bill is 49 dollars. Let x be total units used. Choose the correct branch, solve and check its condition.

First calculate the price of 20 units.
Only x − 20 units use the higher rate.
Subtract 40 before dividing by 3.

49 exceeds the first-block bill of 40, so use x > 20.
40 + 3(x − 20) = 49 gives x − 20 = 3 and x = 23.
Check the bill and the condition 23 > 20.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Further practice11

Two collections total 26 items. One has 6 more than the other. Let x be the smaller count. Find both.

Express both counts using x.
The larger is x + 6.
Use their total 26.

x + (x + 6) = 26 gives 2x = 20.
x = 10; check 10 + 16 = 26 and 16 − 10 = 6.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Further practice12

A rectangle is 3 cm longer than it is wide and has perimeter 30 cm. Let x be its width. Find the dimensions.

Use perimeter, not area.
The length is x + 3.
Count both opposite sides.

There are two widths and two lengths: 2x + 2(x + 3) = 30.
4x + 6 = 30 gives x = 6.
Check 2 × 6 + 2 × 9 = 30.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Further practice13

A slower runner is 60 m ahead. Speeds remain 3 m/s and 5 m/s in the same direction. Let x be the faster runner’s catching time in seconds. Find x.

For same-direction catching, use the speed difference.
The gap to close is 60 m.
Divide 60 by 2.

The gap closes at 5 − 3 = 2 m/s.
2x = 60 gives x = 30.
Check the faster distance exceeds the slower distance by 60 m.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Further practice14

Two workers take 3 hours and 6 hours alone for the same job. At constant additive rates, how long do they take together? Let the time be x hours.

Use the completed-work fractions.
The total completed work is 1.
The combined rate is 1/2 job per hour.

x/3 + x/6 = 1.
Multiply by 6: 2x + x = 6, so x = 2.
Check 2/3 + 2/6 = 1.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Extension15

Eight tickets cost 200 dollars. Adult tickets cost 30 dollars and student tickets 20 dollars. Let x be the adult-ticket count. Find both counts and check that they are possible.

Do not give the two ticket types independent unknowns.
Use the total 8 to write the second count.
Check both total count and total price.

The student-ticket count is 8 − x.
30x + 20(8 − x) = 200 gives 10x = 40 and x = 4.
Both counts are whole numbers between 0 and 8; cost is 120 + 80 = 200.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Extension16

Use the same tariff: first 20 units at 2 dollars, extra units at 3 dollars. The bill is only 30 dollars. Find usage x and explain why the extra-unit formula would use the wrong branch.

Compare 30 with the 40-dollar boundary.
All used units are within the first block.
Use 2x = 30 and check the range.

The first-block maximum bill is 20 × 2 = 40 dollars; 30 is below it.
Use 2x = 30, giving x = 15.
Check 15 ≤ 20 and 2 × 15 = 30.

Check your work

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Preparation and assessment notes

Prerequisites

Linear equations including brackets and constant fractions; speed × time and percentage bases.

Look for in students’ work

Use the prerequisite checks to choose support, project the worked and guided items, then let learners work on paper. Select supplements or extensions as needed. No learner answers are saved by this presentation.

Collect the paper exit questions and note completion and common errors. This page does not grade or store paper answers.

Paper for the class

Download all worksheets (ZIP · 318 PDFs · 47.2 MB)

All 53 Year 7–8 upper- and lower-volume packs in Traditional Chinese and English. Includes teacher answers; student core, further practice and teacher papers are in separate folders.

Core student worksheet: two A4 pages. Further practice is separate. Teacher copies include reference reasoning.

Print edition

Textbook reference and authorship

A 40-minute selection of original core practice after studying the cited textbook lessons. It is not a replacement for every lesson in the chapter.

大湾区初中数学三维课堂（七年级上册）正文.pdf

Year 7, upper volume · text pp. 67–78 · foundation lessons 34–39

Original questions and explanations; the source textbook and answer PDFs are not reproduced here.

Equations in context: quantities, rates and conditions

Year 7, upper volume · text pp. 67–78 · foundation lessons 34–39 · 1/2

Name: __________ Class: ______ Date: __________

Worked example1

Two groups collected 32 cards in total. The larger group collected 8 more than the smaller. Let x be the smaller group’s number. Find both numbers.

Guided attempt2

A walker moves at 60 m/min. A runner starts from the same point 2 minutes later at 80 m/min in the same direction. Speeds stay constant. Let x be the runner’s time until catching up. Find x.

Walker: time ____ min; distance ____ m.
Runner: time ____ min; distance ____ m.
Equal-distance equation: ____.

Prerequisite3

One worker completes 1/6 of a job each hour and another completes 1/3. Find their combined hourly rate.

\frac{1}{6}+\frac{1}{3}

Prerequisite4

Profit is 20% of a 50-dollar cost. Find the profit, not the selling price.

\frac{20}{100}\times50

Equations in context: quantities, rates and conditions

Year 7, upper volume · text pp. 67–78 · foundation lessons 34–39 · 2/2

Name: __________ Class: ______ Date: __________

Independent practice5

Notebooks cost 4 dollars each, plus a single 3-dollar packing fee. The total is 27 dollars. Let x be the number of notebooks. Find x.

Independent practice6

Two workers would finish the same job alone in 4 hours and 6 hours. They work together at constant independent rates. Let x be their time together. Find x.

Independent practice7

A product sells for 60 dollars at a profit of 20% of its cost. Let the cost be x dollars. Find x.

Independent practice8

The ratio of boys to girls is 2:3 and there are 35 pupils. Let x be the number of boys. Find x and check both counts.

Exit question9

An item marked at 80 dollars is sold at 25% off. That selling price is 20% above cost. Let cost be x dollars. Find x and identify both percentage bases.

Exit question10

The first 20 units cost 2 dollars each; additional units cost 3 dollars each. A bill is 49 dollars. Let x be total units used. Choose the correct branch, solve and check its condition.

Equations in context: quantities, rates and conditions — Further practice

Year 7, upper volume · text pp. 67–78 · foundation lessons 34–39

Further practice11

Two collections total 26 items. One has 6 more than the other. Let x be the smaller count. Find both.

Further practice12

A rectangle is 3 cm longer than it is wide and has perimeter 30 cm. Let x be its width. Find the dimensions.

Further practice13

A slower runner is 60 m ahead. Speeds remain 3 m/s and 5 m/s in the same direction. Let x be the faster runner’s catching time in seconds. Find x.

Further practice14

Two workers take 3 hours and 6 hours alone for the same job. At constant additive rates, how long do they take together? Let the time be x hours.

Extension15

Eight tickets cost 200 dollars. Adult tickets cost 30 dollars and student tickets 20 dollars. Let x be the adult-ticket count. Find both counts and check that they are possible.

Extension16

Use the same tariff: first 20 units at 2 dollars, extra units at 3 dollars. The bill is only 30 dollars. Find usage x and explain why the extra-unit formula would use the wrong branch.

Equations in context: quantities, rates and conditions — Teacher answers

Year 7, upper volume · text pp. 67–78 · foundation lessons 34–39

Worked example1

Two groups collected 32 cards in total. The larger group collected 8 more than the smaller. Let x be the smaller group’s number. Find both numbers.

Reference answer

x = 12 cards; the larger group has 20 cards.

x=12

The larger number is x + 8, so x + (x + 8) = 32.
2x + 8 = 32 gives x = 12; the other number is 20.
Check: 12 + 20 = 32 and 20 − 12 = 8.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Guided attempt2

A walker moves at 60 m/min. A runner starts from the same point 2 minutes later at 80 m/min in the same direction. Speeds stay constant. Let x be the runner’s time until catching up. Find x.

Walker: time ____ min; distance ____ m.
Runner: time ____ min; distance ____ m.
Equal-distance equation: ____.

Reference answer

x = 6 minutes. The walker travels for 8 minutes; both cover 480 m.

x=6

The walker’s time is x + 2, whereas the runner’s time is x.
Equal distances give 60(x + 2) = 80x.
60x + 120 = 80x gives x = 6; check 60 × 8 = 80 × 6 = 480.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Prerequisite3

One worker completes 1/6 of a job each hour and another completes 1/3. Find their combined hourly rate.

\frac{1}{6}+\frac{1}{3}

Reference answer

1/2 of the job per hour.

\frac{1}{2}

The rates use the same whole job and the same hour unit.
1/6 + 1/3 = 1/6 + 2/6 = 1/2.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Prerequisite4

Profit is 20% of a 50-dollar cost. Find the profit, not the selling price.

\frac{20}{100}\times50

Reference answer

10 dollars of profit; the cost is the percentage base.

10

20% means 20/100 of the cost.
Multiply by 50 to get 10.

Apply the relevant sign and order rules to every operation.
Show an intermediate calculation and obtain the exact stated value.

Independent practice5

Notebooks cost 4 dollars each, plus a single 3-dollar packing fee. The total is 27 dollars. Let x be the number of notebooks. Find x.

Reference answer

6 notebooks.

x=6

4x + 3 = 27.
4x = 24 gives x = 6. Check 4 × 6 + 3 = 27.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Independent practice6

Two workers would finish the same job alone in 4 hours and 6 hours. They work together at constant independent rates. Let x be their time together. Find x.

Reference answer

12/5 hours, or 2.4 hours.

x=\frac{12}{5}

Each contribution is rate × time: x/4 and x/6; their sum is one job.
Multiply x/4 + x/6 = 1 by 12: 3x + 2x = 12, so x = 12/5.
Check: 3/5 + 2/5 = 1, and the time is shorter than either solo time.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Independent practice7

A product sells for 60 dollars at a profit of 20% of its cost. Let the cost be x dollars. Find x.

Reference answer

The cost is 50 dollars.

x=50

Selling price = cost + 20% of cost = 1.2x.
1.2x = 60 gives x = 50. Check 50 + 10 = 60.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Independent practice8

The ratio of boys to girls is 2:3 and there are 35 pupils. Let x be the number of boys. Find x and check both counts.

Reference answer

14 boys and 21 girls.

x=14

Girls number (3/2)x, so x + (3/2)x = 35.
(5/2)x = 35 gives x = 14; girls = 21.
Check 14 + 21 = 35 and 14:21 = 2:3.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Exit question9

An item marked at 80 dollars is sold at 25% off. That selling price is 20% above cost. Let cost be x dollars. Find x and identify both percentage bases.

Reference answer

Cost = 50 dollars. Discount uses the marked price; profit uses the cost.

x=50

Selling price = 80 × 75% = 60 dollars.
60 = 1.2x, so x = 50.
Check 50 × 120% = 60 and 80 × 75% = 60.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Exit question10

The first 20 units cost 2 dollars each; additional units cost 3 dollars each. A bill is 49 dollars. Let x be total units used. Choose the correct branch, solve and check its condition.

Reference answer

23 units. The first 20 cost 40 dollars and the extra 3 cost 9 dollars.

x=23

49 exceeds the first-block bill of 40, so use x > 20.
40 + 3(x − 20) = 49 gives x − 20 = 3 and x = 23.
Check the bill and the condition 23 > 20.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Further practice11

Two collections total 26 items. One has 6 more than the other. Let x be the smaller count. Find both.

Reference answer

10 items and 16 items.

x=10

x + (x + 6) = 26 gives 2x = 20.
x = 10; check 10 + 16 = 26 and 16 − 10 = 6.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Further practice12

A rectangle is 3 cm longer than it is wide and has perimeter 30 cm. Let x be its width. Find the dimensions.

Reference answer

Width 6 cm and length 9 cm.

x=6

There are two widths and two lengths: 2x + 2(x + 3) = 30.
4x + 6 = 30 gives x = 6.
Check 2 × 6 + 2 × 9 = 30.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Further practice13

A slower runner is 60 m ahead. Speeds remain 3 m/s and 5 m/s in the same direction. Let x be the faster runner’s catching time in seconds. Find x.

Reference answer

30 seconds.

x=30

The gap closes at 5 − 3 = 2 m/s.
2x = 60 gives x = 30.
Check the faster distance exceeds the slower distance by 60 m.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Further practice14

Two workers take 3 hours and 6 hours alone for the same job. At constant additive rates, how long do they take together? Let the time be x hours.

Reference answer

2 hours.

x=2

x/3 + x/6 = 1.
Multiply by 6: 2x + x = 6, so x = 2.
Check 2/3 + 2/6 = 1.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Extension15

Eight tickets cost 200 dollars. Adult tickets cost 30 dollars and student tickets 20 dollars. Let x be the adult-ticket count. Find both counts and check that they are possible.

Reference answer

4 adult tickets and 4 student tickets.

x=4

The student-ticket count is 8 − x.
30x + 20(8 − x) = 200 gives 10x = 40 and x = 4.
Both counts are whole numbers between 0 and 8; cost is 120 + 80 = 200.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.

Extension16

Use the same tariff: first 20 units at 2 dollars, extra units at 3 dollars. The bill is only 30 dollars. Find usage x and explain why the extra-unit formula would use the wrong branch.

Reference answer

15 units, within 0 ≤ x ≤ 20. The extra-unit formula assumes x > 20, which is not satisfied.

x=15

The first-block maximum bill is 20 × 2 = 40 dollars; 30 is below it.
Use 2x = 30, giving x = 15.
Check 15 ≤ 20 and 2 × 15 = 30.

Define x with its unit and write an equality matching the situation.
Solve, then check the original quantities, units and any range or whole-number condition.