Negative numbers and linear equations · applications

Applications: Negative Numbers and Linear Equations

10 question types · Secondary 1 · each worked step by step with a figure that follows the steps

PSLE · O-Level · SAT · GCSE Higher

01

Temperature Readings on Either Side of Zero

methodWalk Along the Number Line, Then Add the Signed Readings and Divide

At a mountain hut the temperature at 6 a.m. was −6 °C. By noon it had risen by 10 °C. By 6 p.m. it had fallen by 7 °C from the noon reading, and by midnight it had fallen by a further 8 °C. (a) Find the temperature at midnight. (b) Find the mean of the four readings.

0−6+104
A rise of 10 °C is a move of 10 to the right: −6 + 10 = 4 °C at noon.
Start at −6 on the number line. A rise of 10 °C is a move of 10 to the right: −6 + 10 = 4. The noon reading is 4 °C.
step 1 of 5

A rise is an addition and a fall is a subtraction, so each reading is one move along the number line from the reading before it. The mean is the sum of the four signed readings divided by 4.

  1. Start at −6 on the number line. A rise of 10 °C is a move of 10 to the right: −6 + 10 = 4. The noon reading is 4 °C.
  2. A fall of 7 °C is a move of 7 to the left. From 4 it takes 4 to reach zero and 3 more to pass it: 4 − 7 = −3. The 6 p.m. reading is −3 °C.
  3. A further fall of 8 °C gives −3 − 8 = −11. (a) The temperature at midnight is −11 °C.
  4. Add the four readings. The one positive reading is 4, and the negative readings add up to −6 + (−3) + (−11) = −20. The sum is 4 + (−20) = −16.
  5. (b) The mean is −16 ÷ 4 = −4 °C, because a negative number divided by a positive number is negative. Check: −4 lies between the lowest reading, −11, and the highest, 4.

answer(a) −11 °C; (b) −4 °C

techniqueAdding and Subtracting Across Zero · Multiplying and Dividing Negative Numbers · Below Zero

examsO-Level · GCSE Higher

Common pitfalls

  • Writing −3 − 8 = 5 or −3 − 8 = −5. The temperature is already below zero and it falls again, so the move is further to the left: −3 − 8 = −11.
  • Finding the mean of 6, 4, 3 and 11 and ignoring the signs. The signs are part of the readings, so the sum is −16, not 24.
02

A Game Score with a Penalty Taken Away

methodSubtracting a Negative Number Is Adding Its Opposite

In a card game each card either adds points to a player's score or takes points away. Mei has four cards: +15, −20, +8 and −12. (a) Find Mei's score. (b) The −12 card was dealt to her by mistake, so it is taken away from her. Find her new score.

+15−20+8−12gains: 15 + 8 = 23
The two gains add up to 15 + 8 = 23.
Add the two gains first: 15 + 8 = 23.
step 1 of 5

The score is the sum of the four signed numbers. Taking a card away subtracts its value from the score, and the value of this card is negative.

  1. Add the two gains first: 15 + 8 = 23.
  2. Add the two penalties: (−20) + (−12) = −32.
  3. (a) Mei's score is 23 + (−32) = 23 − 32 = −9 points.
  4. Taking the −12 card away subtracts −12 from her score, so her new score is −9 − (−12).
  5. Subtracting a negative number is the same as adding its opposite: −9 − (−12) = −9 + 12 = 3. (b) Her new score is 3 points. Check: the three cards that remain give 15 − 20 + 8 = 3.

answer(a) −9 points; (b) 3 points

techniqueSubtracting Negative Numbers · Adding and Subtracting Across Zero

examsO-Level

Common pitfalls

  • Working out −9 − 12 = −21 for part (b). Losing a penalty makes the score better, not worse. The card is worth −12, so taking it away is −9 − (−12) = −9 + 12.
  • Writing 23 − 32 = 9 in part (a). The penalties are larger than the gains, so the score is below zero: 23 − 32 = −9.
03

Heights Above and Depths Below Sea Level

methodDistance Is the Absolute Value of the Difference

The lamp of a lighthouse is 45 m above sea level. Directly below it, a diver is at −18 m and a wreck lies on the sea bed at −62 m, where a negative height is a depth below sea level. (a) Find the vertical distance between the lamp and the diver. (b) How much further must the diver descend to reach the wreck?

sea level 0lamp 45 mdiver −18 mwreck −62 m
Sea level is 0. The lamp is at 45, the diver at −18 and the wreck at −62.
Write the heights as signed numbers on a vertical number line: the lamp is at 45, sea level is 0, the diver is at −18 and the wreck is at −62.
step 1 of 5

Put the three heights on a vertical number line with sea level at 0. The distance between two heights is the absolute value of their difference, so it is never negative.

  1. Write the heights as signed numbers on a vertical number line: the lamp is at 45, sea level is 0, the diver is at −18 and the wreck is at −62.
  2. The distance between the lamp and the diver is |45 − (−18)|. Subtracting −18 is adding 18, so this is |45 + 18| = 63.
  3. (a) The lamp and the diver are 63 m apart. On the number line this is 45 m down to sea level and 18 m more below it.
  4. The distance between the diver and the wreck is |−18 − (−62)| = |−18 + 62| = |44| = 44.
  5. (b) The diver must descend 44 m further. Check: 63 + 44 = 107, and the lamp is 45 − (−62) = 107 m above the wreck.

answer(a) 63 m; (b) 44 m

techniqueAbsolute Value · Subtracting Negative Numbers

examsH2

Common pitfalls

  • Subtracting the numbers without their signs in part (a), 45 − 18 = 27. The lamp and the diver are on opposite sides of sea level, so the two distances from zero are added: 45 + 18 = 63.
  • Adding in part (b), 18 + 62 = 80. The diver and the wreck are both below sea level, so the distance between them is the difference of the two depths: 62 − 18 = 44.
04

A Quiz That Takes Marks Away for Wrong Answers

methodSigned Marks, Then a Linear Equation in the Number of Right Answers

A quiz has 20 questions. Each right answer scores 4 marks and each wrong answer scores −2 marks. Aisha and Ben both answer every question. (a) Aisha gets 13 answers right. Find her score. (b) Ben's score is 26. How many answers did Ben get right?

13 right444444444444413 × 4 = 527 wrong−2−2−2−2−2−2−27 × (−2) = −14
Aisha has 13 right answers and 7 wrong answers: 13 × 4 = 52 and 7 × (−2) = −14.
Aisha has 13 right answers and 20 − 13 = 7 wrong answers. The right answers score 13 × 4 = 52 and the wrong answers score 7 × (−2) = −14.
step 1 of 6

A score is 4 times the number of right answers plus −2 times the number of wrong answers. For Ben the number of right answers is unknown, so it is given a letter and the score becomes an equation.

  1. Aisha has 13 right answers and 20 − 13 = 7 wrong answers. The right answers score 13 × 4 = 52 and the wrong answers score 7 × (−2) = −14.
  2. (a) Aisha's score is 52 + (−14) = 38 marks.
  3. Let r be the number of answers Ben got right. Then he got 20 − r wrong, and his score is 4r − 2(20 − r). His score is 26, so 4r − 2(20 − r) = 26.
  4. Expand the bracket: −2 × 20 = −40 and −2 × (−r) = +2r, so 4r − 40 + 2r = 26. Collect the r terms: 6r − 40 = 26.
  5. Add 40 to both sides: 6r = 66. Divide both sides by 6: r = 11.
  6. (b) Ben got 11 answers right. Check: 11 right answers score 44, the 9 wrong answers score −18, and 44 − 18 = 26.

answer(a) 38 marks; (b) 11 right answers

techniqueSolving Linear Equations · Multiplying and Dividing Negative Numbers · Algebraic Notation

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Expanding −2(20 − r) as −40 − 2r. The product of −2 and −r is +2r, because a negative number times a negative number is positive.
  • Dividing 26 by 4 to count the right answers. The wrong answers take marks away, so the score is not just 4 marks for each right answer.
05

Cold Stores Ordered and Compared with a Target

methodOrder on the Number Line, Then Compare Absolute Differences

A warehouse has five cold stores. Their temperatures are: store A, 3 °C; store B, −8 °C; store C, −16 °C; store D, −5 °C; store E, −13 °C. (a) List the stores from the coldest to the warmest. (b) A delivery of fish must be kept as close to −10 °C as possible. Which store is closest to this target, and by how many degrees does it differ from the target?

03A−8B−16C−5D−13E
The further left a temperature is on the number line, the colder the store.
Mark the five temperatures on a number line. The further left a number is, the smaller it is and the colder the store.
step 1 of 5

On a number line the colder temperature is always further to the left. How close a store is to the target is the absolute value of the difference between its temperature and −10.

  1. Mark the five temperatures on a number line. The further left a number is, the smaller it is and the colder the store.
  2. Among the negative numbers, −16 is furthest from zero on the left, so it is the smallest. Then come −13, −8 and −5. The only positive temperature, 3, is the largest.
  3. (a) From the coldest to the warmest the stores are C (−16), E (−13), B (−8), D (−5), A (3).
  4. The target −10 lies between E and B. Store E differs from it by |−13 − (−10)| = |−3| = 3 and store B differs by |−8 − (−10)| = |2| = 2.
  5. (b) Store B is the closest, 2 °C from the target. Check the others: store D differs by |−5 + 10| = 5, store C by |−16 + 10| = 6 and store A by |3 + 10| = 13.

answer(a) C (−16), E (−13), B (−8), D (−5), A (3); (b) store B, 2 °C from the target

techniqueOrdering Negative Numbers · Absolute Value

examsO-Level · GCSE Higher

Common pitfalls

  • Calling store D the coldest after store A because 5 is less than 8, 13 and 16. For negative numbers the order is reversed: −16 is less than −5, so store C is the coldest.
  • Choosing store E because it is the first store colder than the target. A store may differ from the target on either side, and the differences are 3 for store E and 2 for store B, so store B is closer.
06

A Tank Draining at a Steady Rate

methodA Negative Rate of Change and the Time to Reach Zero

A tank holds 240 liters of water. A valve is opened and the volume of water in the tank changes at a steady rate of −15 liters per minute. (a) Find the volume of water in the tank after 7 minutes. (b) After how many minutes is the tank empty?

The tank starts with 240 liters.Tankleft ?−15−15−15−15−15−15−157 × (−15) = −105
In 7 minutes the volume changes by 7 × (−15) = −105 liters.
In 7 minutes the volume changes by 7 × (−15) = −105 liters. A positive number times a negative number is negative, so this is a loss of 105 liters.
step 1 of 5

A rate of −15 liters per minute means the volume goes down by 15 liters every minute. The change after some minutes is the rate multiplied by the time, and the tank is empty when the volume is 0.

  1. In 7 minutes the volume changes by 7 × (−15) = −105 liters. A positive number times a negative number is negative, so this is a loss of 105 liters.
  2. (a) After 7 minutes the tank holds 240 + (−105) = 135 liters.
  3. Let m be the number of minutes until the tank is empty. After m minutes the volume is 240 − 15m liters, and an empty tank holds 0 liters, so 240 − 15m = 0.
  4. Add 15m to both sides: 240 = 15m. Divide both sides by 15: m = 16.
  5. (b) The tank is empty after 16 minutes. Check: 16 × (−15) = −240 and 240 + (−240) = 0.

answer(a) 135 liters; (b) 16 minutes

techniqueMultiplying and Dividing Negative Numbers · Solving Linear Equations

examsO-Level · GCSE Higher

Common pitfalls

  • Adding 105 liters in part (a). The rate is negative, so the change 7 × (−15) is −105 and the volume goes down to 135 liters.
  • Giving m = −16 after writing −15m = −240. Both sides are divided by −15, and a negative number divided by a negative number is positive, so m = 16. A time after the valve is opened cannot be negative.
07

Consecutive Odd House Numbers with a Known Sum

methodOne Letter for the Smallest Number, Then Form and Solve an Equation

The houses on one side of a street have consecutive odd numbers. The numbers of three houses next to one another add up to 111. (a) Find the three house numbers. (b) Find the sum of the numbers of the next three houses along the same side.

FirstnSecondn2Thirdn4
Odd numbers go up in twos, so the three numbers are n, n + 2 and n + 4.
Let the smallest of the three house numbers be n. Consecutive odd numbers go up in twos, so the other two are n + 2 and n + 4.
step 1 of 6

Consecutive odd numbers go up in twos, so all three numbers can be written with one letter. Their sum gives a linear equation in that letter.

  1. Let the smallest of the three house numbers be n. Consecutive odd numbers go up in twos, so the other two are n + 2 and n + 4.
  2. The three numbers add up to 111: n + (n + 2) + (n + 4) = 111. Collect like terms: 3n + 6 = 111.
  3. Subtract 6 from both sides: 3n = 105. Divide both sides by 3: n = 35.
  4. (a) The house numbers are 35, 37 and 39. Check: 35 + 37 + 39 = 111, and 35 is odd.
  5. The next three houses are 41, 43 and 45. Each is 6 more than the house three places before it, so their sum is 3 × 6 = 18 more than 111.
  6. (b) The sum is 111 + 18 = 129. Check: 41 + 43 + 45 = 129.

answer(a) 35, 37 and 39; (b) 129

techniqueAlgebraic Notation · Solving Linear Equations

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Writing the three numbers as n, n + 1 and n + 2. Those are consecutive whole numbers. Odd numbers are 2 apart, so the numbers are n, n + 2 and n + 4.
  • Stopping at 3n = 111 and forgetting the 2 and the 4. The equation is 3n + 6 = 111, and 6 must be subtracted from both sides before dividing by 3.
08

A Taxi Fare with a Fixed Charge and a Charge per Kilometer

methodFixed Part Plus Rate Times Distance, Solved for the Distance

A taxi company charges a fixed $5 for every trip and then $3 for each kilometer traveled. (a) Find the fare for a trip of 12 km. (b) The fare for another trip is $62. How long is that trip?

Fare$5$3 for each km
A trip of d km costs the fixed $5 and $3 for each kilometer: 5 + 3d dollars.
Let the length of a trip be d km. The fare is the fixed charge plus $3 for each kilometer, which is 5 + 3d dollars.
step 1 of 5

Write the fare for a trip of d km as an expression in d. Part (a) substitutes a distance into it, and part (b) sets it equal to the fare and solves for d.

  1. Let the length of a trip be d km. The fare is the fixed charge plus $3 for each kilometer, which is 5 + 3d dollars.
  2. (a) For d = 12 the fare is 5 + 3 × 12 = 5 + 36 = $41.
  3. For the other trip the fare is $62, so 5 + 3d = 62.
  4. Subtract 5 from both sides: 3d = 57. Divide both sides by 3: d = 19.
  5. (b) The trip is 19 km long. Check: 5 + 3 × 19 = 5 + 57 = 62.

answer(a) $41; (b) 19 km

techniqueSolving Linear Equations · Algebraic Notation

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Dividing the whole fare by 3, 62 ÷ 3. The fixed charge of $5 is not paid per kilometer, so it is subtracted first and only the remaining $57 is divided by 3.
  • Working out (5 + 3) × 12 = 96 in part (a). The fixed charge is paid once, not once for every kilometer, so the fare is 5 + 3 × 12.
09

Two Membership Plans That Cost the Same

methodAn Equation with the Unknown on Both Sides

A swimming pool offers two plans. Plan A costs $20 a month and then $4 for each visit. Plan B costs $44 a month and then $1 for each visit. (a) For how many visits in a month do the two plans cost the same, and what is that cost? (b) Hana swims 15 times a month. Which plan is cheaper for her, and by how much?

Plan A$20v × $4Plan B$44v × $1
For v visits Plan A costs 20 + 4v dollars and Plan B costs 44 + v dollars.
Let v be the number of visits in a month. Plan A costs 20 + 4v dollars and Plan B costs 44 + v dollars.
step 1 of 6

Write the monthly cost of each plan in terms of the number of visits. The plans cost the same when the two expressions are equal, which is an equation with the unknown on both sides.

  1. Let v be the number of visits in a month. Plan A costs 20 + 4v dollars and Plan B costs 44 + v dollars.
  2. The two plans cost the same when 20 + 4v = 44 + v.
  3. Subtract v from both sides: 20 + 3v = 44. Subtract 20 from both sides: 3v = 24. Divide both sides by 3: v = 8.
  4. (a) The plans cost the same for 8 visits. Check: Plan A costs 20 + 4 × 8 = $52 and Plan B costs 44 + 8 = $52.
  5. For 15 visits Plan A costs 20 + 4 × 15 = $80 and Plan B costs 44 + 15 = $59.
  6. (b) Plan B is cheaper by 80 − 59 = $21. After the 8th visit each extra visit adds $4 to Plan A and only $1 to Plan B.

answer(a) 8 visits, $52; (b) Plan B, by $21

techniqueSolving Linear Equations · Algebraic Notation

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Subtracting v from one side only and writing 20 + 4v = 44. Whatever is done to one side must be done to the other, so the equation becomes 20 + 3v = 44.
  • Choosing Plan A for 15 visits because its monthly charge is lower. The charge for each visit matters as well, and beyond 8 visits Plan A costs more.
10

A Rectangle with Sides Written in Terms of x

methodForm the Perimeter Equation, Solve It, Then Substitute

A rectangular picture frame is (2x + 5) cm long and (x − 1) cm wide. A strip of wood exactly 50 cm long goes once round its edge. (a) Find the value of x. (b) Find the area of the rectangle.

(2x + 5) cm(x − 1) cmThe perimeter is 50 cm.2(2x + 5) + 2(x − 1)=50
Twice the length plus twice the width is the perimeter: 2(2x + 5) + 2(x − 1) = 50.
The perimeter is twice the length plus twice the width, and it is 50 cm: 2(2x + 5) + 2(x − 1) = 50.
step 1 of 6

The strip of wood is the perimeter, which is twice the length plus twice the width. That gives a linear equation in x. The value of x then gives the two sides and the area.

  1. The perimeter is twice the length plus twice the width, and it is 50 cm: 2(2x + 5) + 2(x − 1) = 50.
  2. Expand the brackets: 4x + 10 + 2x − 2 = 50. Collect like terms: 6x + 8 = 50.
  3. Subtract 8 from both sides: 6x = 42. Divide both sides by 6: x = 7.
  4. (a) x = 7. The length is 2 × 7 + 5 = 19 cm and the width is 7 − 1 = 6 cm. Both sides are positive lengths, so this value of x fits the rectangle.
  5. The area is the length times the width: 19 × 6 = 114.
  6. (b) The area is 114 cm2. Check: the perimeter is 2 × (19 + 6) = 2 × 25 = 50 cm.

answer(a) x = 7; (b) 114 cm2

techniqueSolving Linear Equations · Algebraic Notation

examsPSLE · O-Level · SAT · GCSE Higher

Common pitfalls

  • Adding one length and one width only, (2x + 5) + (x − 1) = 50. A rectangle has two lengths and two widths, so the sum of one of each is half the perimeter, 25 cm.
  • Expanding 2(x − 1) as 2x − 1. The 2 multiplies both terms inside the bracket, so 2(x − 1) = 2x − 2.
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