Practice · When two quantities are not linked, find the one that sits between them · Sensim Math

Variant 1 easy

A car can travel $22.68\ \text{km}$ on $2.1\ \text{L}$ of petrol. If petrol costs $1210$ won per liter, how much does the petrol cost for a $432\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Show solution

1 · Understandwhat's really being asked

A car does 22.68 km on 2.1 L and petrol is 1210 won a litre. We want the petrol cost of a 432 km trip.

Givens

2.1 L takes the car 22.68 km.
Petrol costs 1210 won per litre.
The trip is 432 km.

Unknowns

The cost of the petrol for the trip.

Constraints

The car covers the same distance on every litre.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Kilometres and won have nothing to do with each other directly. Litres are what both rates are about, so go through litres: first how far one goes, then how many the trip needs, then what they cost.

3 · Execute3 carry out the plan

1Find how far one litre goes

#9 Solve an Easier Related Problem 6.RP.A.2

22.68 km spread over 2.1 L.

22.68 \div 2.1 = 10.8

A litre is worth 10.8 km.

2Find the litres the trip needs

#7 Identify Subproblems 6.NS.B.3

432 km at 10.8 km a litre.

432 \div 10.8 = 40

40 litres.

3Cost those litres

#8 Analyze the Units 6.RP.A.3

Each one is 1210 won.

40 \times 1210 = 48{,}400

48,400 won of petrol.

Answer: 48,400 won

4 · Reviewdoes it hold up?

Checking backwards: 40 litres at 10.8 km each covers 432 km, exactly the trip.

Another way: Working out the cost per kilometre first -- 1210 won divided by 10.8 km -- and multiplying by 432 reaches the same total, going through money per kilometre rather than litres.

Standardsmin grade 6

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading kilometres per litre as a unit rate.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning litres into a cost at a fixed price.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the rate and the litres.

💡Takeaway. If two things in a problem never mention each other, look for the third thing that mentions both.

Variant 2 easy

A car can travel $14.4\ \text{km}$ on $1.2\ \text{L}$ of petrol. If petrol costs $1250$ won per liter, how much does the petrol cost for a $456\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Variant 3 easy

A car can travel $11.52\ \text{km}$ on $1.8\ \text{L}$ of petrol. If petrol costs $1390$ won per liter, how much does the petrol cost for a $384\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Show solution

1 · Understandwhat's really being asked

A car does 11.52 km on 1.8 L and petrol is 1390 won a litre. We want the petrol cost of a 384 km trip.

Givens

1.8 L takes the car 11.52 km.
Petrol costs 1390 won per litre.
The trip is 384 km.

Unknowns

The cost of the petrol for the trip.

Constraints

The car covers the same distance on every litre.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Kilometres and won have nothing to do with each other directly. Litres are what both rates are about, so go through litres: first how far one goes, then how many the trip needs, then what they cost.

3 · Execute3 carry out the plan

1Find how far one litre goes

#9 Solve an Easier Related Problem 6.RP.A.2

11.52 km spread over 1.8 L.

11.52 \div 1.8 = 6.4

A litre is worth 6.4 km.

2Find the litres the trip needs

#7 Identify Subproblems 6.NS.B.3

384 km at 6.4 km a litre.

384 \div 6.4 = 60

60 litres.

3Cost those litres

#8 Analyze the Units 6.RP.A.3

Each one is 1390 won.

60 \times 1390 = 83{,}400

83,400 won of petrol.

Answer: 83,400 won

4 · Reviewdoes it hold up?

Checking backwards: 60 litres at 6.4 km each covers 384 km, exactly the trip.

Another way: Working out the cost per kilometre first -- 1390 won divided by 6.4 km -- and multiplying by 384 reaches the same total, going through money per kilometre rather than litres.

Standardsmin grade 6

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading kilometres per litre as a unit rate.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning litres into a cost at a fixed price.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the rate and the litres.

💡Takeaway. If two things in a problem never mention each other, look for the third thing that mentions both.

Variant 4 easy

A car can travel $28.14\ \text{km}$ on $2.1\ \text{L}$ of petrol. If petrol costs $1430$ won per liter, how much does the petrol cost for a $335\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Show solution

1 · Understandwhat's really being asked

A car does 28.14 km on 2.1 L and petrol is 1430 won a litre. We want the petrol cost of a 335 km trip.

Givens

2.1 L takes the car 28.14 km.
Petrol costs 1430 won per litre.
The trip is 335 km.

Unknowns

The cost of the petrol for the trip.

Constraints

The car covers the same distance on every litre.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Kilometres and won have nothing to do with each other directly. Litres are what both rates are about, so go through litres: first how far one goes, then how many the trip needs, then what they cost.

3 · Execute3 carry out the plan

1Find how far one litre goes

#9 Solve an Easier Related Problem 6.RP.A.2

28.14 km spread over 2.1 L.

28.14 \div 2.1 = 13.4

A litre is worth 13.4 km.

2Find the litres the trip needs

#7 Identify Subproblems 6.NS.B.3

335 km at 13.4 km a litre.

335 \div 13.4 = 25

25 litres.

3Cost those litres

#8 Analyze the Units 6.RP.A.3

Each one is 1430 won.

25 \times 1430 = 35{,}750

35,750 won of petrol.

Answer: 35,750 won

4 · Reviewdoes it hold up?

Checking backwards: 25 litres at 13.4 km each covers 335 km, exactly the trip.

Another way: Working out the cost per kilometre first -- 1430 won divided by 13.4 km -- and multiplying by 335 reaches the same total, going through money per kilometre rather than litres.

Standardsmin grade 6

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading kilometres per litre as a unit rate.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning litres into a cost at a fixed price.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the rate and the litres.

💡Takeaway. If two things in a problem never mention each other, look for the third thing that mentions both.

Variant 5 medium

A car can travel $42.84\ \text{km}$ on $1.8\ \text{L}$ of petrol. If petrol costs $1470$ won per liter, how much does the petrol cost for a $357\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Show solution

1 · Understandwhat's really being asked

A car does 42.84 km on 1.8 L and petrol is 1470 won a litre. We want the petrol cost of a 357 km trip.

Givens

1.8 L takes the car 42.84 km.
Petrol costs 1470 won per litre.
The trip is 357 km.

Unknowns

The cost of the petrol for the trip.

Constraints

The car covers the same distance on every litre.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Kilometres and won have nothing to do with each other directly. Litres are what both rates are about, so go through litres: first how far one goes, then how many the trip needs, then what they cost.

3 · Execute3 carry out the plan

1Find how far one litre goes

#9 Solve an Easier Related Problem 6.RP.A.2

42.84 km spread over 1.8 L.

42.84 \div 1.8 = 23.8

A litre is worth 23.8 km.

2Find the litres the trip needs

#7 Identify Subproblems 6.NS.B.3

357 km at 23.8 km a litre.

357 \div 23.8 = 15

15 litres.

3Cost those litres

#8 Analyze the Units 6.RP.A.3

Each one is 1470 won.

15 \times 1470 = 22{,}050

22,050 won of petrol.

Answer: 22,050 won

4 · Reviewdoes it hold up?

Checking backwards: 15 litres at 23.8 km each covers 357 km, exactly the trip.

Another way: Working out the cost per kilometre first -- 1470 won divided by 23.8 km -- and multiplying by 357 reaches the same total, going through money per kilometre rather than litres.

Standardsmin grade 6

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading kilometres per litre as a unit rate.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning litres into a cost at a fixed price.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the rate and the litres.

💡Takeaway. If two things in a problem never mention each other, look for the third thing that mentions both.

Variant 6 medium

A car can travel $30\ \text{km}$ on $1.5\ \text{L}$ of petrol. If petrol costs $1550$ won per liter, how much does the petrol cost for a $740\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Variant 7 medium

A car can travel $12\ \text{km}$ on $1.2\ \text{L}$ of petrol. If petrol costs $1640$ won per liter, how much does the petrol cost for a $480\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Variant 8 medium

A car can travel $66.5\ \text{km}$ on $3.5\ \text{L}$ of petrol. If petrol costs $1640$ won per liter, how much does the petrol cost for a $1121\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Show solution

1 · Understandwhat's really being asked

A car does 66.5 km on 3.5 L and petrol is 1640 won a litre. We want the petrol cost of a 1121 km trip.

Givens

3.5 L takes the car 66.5 km.
Petrol costs 1640 won per litre.
The trip is 1121 km.

Unknowns

The cost of the petrol for the trip.

Constraints

The car covers the same distance on every litre.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Kilometres and won have nothing to do with each other directly. Litres are what both rates are about, so go through litres: first how far one goes, then how many the trip needs, then what they cost.

3 · Execute3 carry out the plan

1Find how far one litre goes

#9 Solve an Easier Related Problem 6.RP.A.2

66.5 km spread over 3.5 L.

66.5 \div 3.5 = 19

A litre is worth 19 km.

2Find the litres the trip needs

#7 Identify Subproblems 6.NS.B.3

1121 km at 19 km a litre.

1121 \div 19 = 59

59 litres.

3Cost those litres

#8 Analyze the Units 6.RP.A.3

Each one is 1640 won.

59 \times 1640 = 96{,}760

96,760 won of petrol.

Answer: 96,760 won

4 · Reviewdoes it hold up?

Checking backwards: 59 litres at 19 km each covers 1121 km, exactly the trip.

Another way: Working out the cost per kilometre first -- 1640 won divided by 19 km -- and multiplying by 1121 reaches the same total, going through money per kilometre rather than litres.

Standardsmin grade 6

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading kilometres per litre as a unit rate.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning litres into a cost at a fixed price.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the rate and the litres.

💡Takeaway. If two things in a problem never mention each other, look for the third thing that mentions both.

Variant 9 hard

A car can travel $38.5\ \text{km}$ on $2.5\ \text{L}$ of petrol. If petrol costs $1680$ won per liter, how much does the petrol cost for a $693\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Show solution

1 · Understandwhat's really being asked

A car does 38.5 km on 2.5 L and petrol is 1680 won a litre. We want the petrol cost of a 693 km trip.

Givens

2.5 L takes the car 38.5 km.
Petrol costs 1680 won per litre.
The trip is 693 km.

Unknowns

The cost of the petrol for the trip.

Constraints

The car covers the same distance on every litre.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Kilometres and won have nothing to do with each other directly. Litres are what both rates are about, so go through litres: first how far one goes, then how many the trip needs, then what they cost.

3 · Execute3 carry out the plan

1Find how far one litre goes

#9 Solve an Easier Related Problem 6.RP.A.2

38.5 km spread over 2.5 L.

38.5 \div 2.5 = 15.4

A litre is worth 15.4 km.

2Find the litres the trip needs

#7 Identify Subproblems 6.NS.B.3

693 km at 15.4 km a litre.

693 \div 15.4 = 45

45 litres.

3Cost those litres

#8 Analyze the Units 6.RP.A.3

Each one is 1680 won.

45 \times 1680 = 75{,}600

75,600 won of petrol.

Answer: 75,600 won

4 · Reviewdoes it hold up?

Checking backwards: 45 litres at 15.4 km each covers 693 km, exactly the trip.

Another way: Working out the cost per kilometre first -- 1680 won divided by 15.4 km -- and multiplying by 693 reaches the same total, going through money per kilometre rather than litres.

Standardsmin grade 6

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading kilometres per litre as a unit rate.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning litres into a cost at a fixed price.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the rate and the litres.

💡Takeaway. If two things in a problem never mention each other, look for the third thing that mentions both.

Variant 10 hard

A car can travel $30.24\ \text{km}$ on $1.8\ \text{L}$ of petrol. If petrol costs $1700$ won per liter, how much does the petrol cost for a $168\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Show solution

1 · Understandwhat's really being asked

A car does 30.24 km on 1.8 L and petrol is 1700 won a litre. We want the petrol cost of a 168 km trip.

Givens

1.8 L takes the car 30.24 km.
Petrol costs 1700 won per litre.
The trip is 168 km.

Unknowns

The cost of the petrol for the trip.

Constraints

The car covers the same distance on every litre.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Kilometres and won have nothing to do with each other directly. Litres are what both rates are about, so go through litres: first how far one goes, then how many the trip needs, then what they cost.

3 · Execute3 carry out the plan

1Find how far one litre goes

#9 Solve an Easier Related Problem 6.RP.A.2

30.24 km spread over 1.8 L.

30.24 \div 1.8 = 16.8

A litre is worth 16.8 km.

2Find the litres the trip needs

#7 Identify Subproblems 6.NS.B.3

168 km at 16.8 km a litre.

168 \div 16.8 = 10

10 litres.

3Cost those litres

#8 Analyze the Units 6.RP.A.3

Each one is 1700 won.

10 \times 1700 = 17{,}000

17,000 won of petrol.

Answer: 17,000 won

4 · Reviewdoes it hold up?

Checking backwards: 10 litres at 16.8 km each covers 168 km, exactly the trip.

Another way: Working out the cost per kilometre first -- 1700 won divided by 16.8 km -- and multiplying by 168 reaches the same total, going through money per kilometre rather than litres.

Standardsmin grade 6

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading kilometres per litre as a unit rate.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning litres into a cost at a fixed price.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the rate and the litres.

💡Takeaway. If two things in a problem never mention each other, look for the third thing that mentions both.

Variant 11 hard

A car can travel $39\ \text{km}$ on $3\ \text{L}$ of petrol. If petrol costs $1730$ won per liter, how much does the petrol cost for a $312\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Variant 12 hard

A car can travel $46.2\ \text{km}$ on $2.1\ \text{L}$ of petrol. If petrol costs $1770$ won per liter, how much does the petrol cost for a $1210\ \text{km}$ trip in this car? (The car covers the same distance on every liter.)

Your answer

Show solution

1 · Understandwhat's really being asked

A car does 46.2 km on 2.1 L and petrol is 1770 won a litre. We want the petrol cost of a 1210 km trip.

Givens

2.1 L takes the car 46.2 km.
Petrol costs 1770 won per litre.
The trip is 1210 km.

Unknowns

The cost of the petrol for the trip.

Constraints

The car covers the same distance on every litre.

2 · Planchoose the strategy

#8 Analyze the Units · also uses: #7 Identify Subproblems#9 Solve an Easier Related Problem

Kilometres and won have nothing to do with each other directly. Litres are what both rates are about, so go through litres: first how far one goes, then how many the trip needs, then what they cost.

3 · Execute3 carry out the plan

1Find how far one litre goes

#9 Solve an Easier Related Problem 6.RP.A.2

46.2 km spread over 2.1 L.

46.2 \div 2.1 = 22

A litre is worth 22 km.

2Find the litres the trip needs

#7 Identify Subproblems 6.NS.B.3

1210 km at 22 km a litre.

1210 \div 22 = 55

55 litres.

3Cost those litres

#8 Analyze the Units 6.RP.A.3

Each one is 1770 won.

55 \times 1770 = 97{,}350

97,350 won of petrol.

Answer: 97,350 won

4 · Reviewdoes it hold up?

Checking backwards: 55 litres at 22 km each covers 1210 km, exactly the trip.

Another way: Working out the cost per kilometre first -- 1770 won divided by 22 km -- and multiplying by 1210 reaches the same total, going through money per kilometre rather than litres.

Standardsmin grade 6

6.RP.A.2 Understand unit rate a/b associated with a ratio — Reading kilometres per litre as a unit rate.
6.RP.A.3 Use ratio and rate reasoning to solve problems — Turning litres into a cost at a fixed price.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing decimals to get the rate and the litres.

💡Takeaway. If two things in a problem never mention each other, look for the third thing that mentions both.

When two quantities are not linked, find the one that sits between them · 12 practice problems

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