← Add the two facts and one unknown cancels itself · Find Two Unknowns from Sum and Difference

Add the two facts and one unknown cancels itself · 12 practice problems

6.EE.B.76.EE.B.66.NS.B.3

Generated variants — 12

Freshly produced from the archetype’s parameters — problem, figure, and solution derived together.

Variant 1 easy answer: 4.6

Two numbers have a sum of $30.8$ and a difference of $19.8$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 30.8 and differ by 19.8. We want the bigger one divided by the smaller.

Givens

Their sum is 30.8.
Their difference is 19.8.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

30.8 + 19.8 = 50.6 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

50.6 \div 2 = 25.3

The larger is 25.3.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

30.8 - 25.3 = 5.5

The smaller is 5.5.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

25.3 \div 5.5 = 4.6

The quotient is 4.6.

Answer: 4.6

4 · Reviewdoes it hold up?

25.3 and 5.5 add to 30.8 and differ by 19.8, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 11, so 5.5 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 2 easy answer: 19.8

Two numbers have a sum of $31.2$ and a difference of $28.2$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 31.2 and differ by 28.2. We want the bigger one divided by the smaller.

Givens

Their sum is 31.2.
Their difference is 28.2.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

31.2 + 28.2 = 59.4 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

59.4 \div 2 = 29.7

The larger is 29.7.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

31.2 - 29.7 = 1.5

The smaller is 1.5.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

29.7 \div 1.5 = 19.8

The quotient is 19.8.

Answer: 19.8

4 · Reviewdoes it hold up?

29.7 and 1.5 add to 31.2 and differ by 28.2, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 3, so 1.5 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 3 easy answer: 24.75

Two numbers have a sum of $41.2$ and a difference of $38$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 41.2 and differ by 38. We want the bigger one divided by the smaller.

Givens

Their sum is 41.2.
Their difference is 38.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

41.2 + 38 = 79.2 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

79.2 \div 2 = 39.6

The larger is 39.6.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

41.2 - 39.6 = 1.6

The smaller is 1.6.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

39.6 \div 1.6 = 24.75

The quotient is 24.75.

Answer: 24.75

4 · Reviewdoes it hold up?

39.6 and 1.6 add to 41.2 and differ by 38, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 3.2, so 1.6 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 4 easy answer: 4.65

Two numbers have a sum of $45.2$ and a difference of $29.2$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 45.2 and differ by 29.2. We want the bigger one divided by the smaller.

Givens

Their sum is 45.2.
Their difference is 29.2.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

45.2 + 29.2 = 74.4 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

74.4 \div 2 = 37.2

The larger is 37.2.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

45.2 - 37.2 = 8

The smaller is 8.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

37.2 \div 8 = 4.65

The quotient is 4.65.

Answer: 4.65

4 · Reviewdoes it hold up?

37.2 and 8 add to 45.2 and differ by 29.2, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 16, so 8 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 5 medium answer: 12.35

Two numbers have a sum of $53.4$ and a difference of $45.4$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 53.4 and differ by 45.4. We want the bigger one divided by the smaller.

Givens

Their sum is 53.4.
Their difference is 45.4.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

53.4 + 45.4 = 98.8 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

98.8 \div 2 = 49.4

The larger is 49.4.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

53.4 - 49.4 = 4

The smaller is 4.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

49.4 \div 4 = 12.35

The quotient is 12.35.

Answer: 12.35

4 · Reviewdoes it hold up?

49.4 and 4 add to 53.4 and differ by 45.4, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 8, so 4 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 6 medium answer: 62.2

Two numbers have a sum of $63.2$ and a difference of $61.2$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 63.2 and differ by 61.2. We want the bigger one divided by the smaller.

Givens

Their sum is 63.2.
Their difference is 61.2.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

63.2 + 61.2 = 124.4 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

124.4 \div 2 = 62.2

The larger is 62.2.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

63.2 - 62.2 = 1

The smaller is 1.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

62.2 \div 1 = 62.2

The quotient is 62.2.

Answer: 62.2

4 · Reviewdoes it hold up?

62.2 and 1 add to 63.2 and differ by 61.2, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 2, so 1 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 7 medium answer: 12.1

Two numbers have a sum of $65.5$ and a difference of $55.5$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 65.5 and differ by 55.5. We want the bigger one divided by the smaller.

Givens

Their sum is 65.5.
Their difference is 55.5.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

65.5 + 55.5 = 121 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

121 \div 2 = 60.5

The larger is 60.5.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

65.5 - 60.5 = 5

The smaller is 5.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

60.5 \div 5 = 12.1

The quotient is 12.1.

Answer: 12.1

4 · Reviewdoes it hold up?

60.5 and 5 add to 65.5 and differ by 55.5, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 10, so 5 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 8 medium answer: 29.64

Two numbers have a sum of $76.6$ and a difference of $71.6$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 76.6 and differ by 71.6. We want the bigger one divided by the smaller.

Givens

Their sum is 76.6.
Their difference is 71.6.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

76.6 + 71.6 = 148.2 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

148.2 \div 2 = 74.1

The larger is 74.1.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

76.6 - 74.1 = 2.5

The smaller is 2.5.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

74.1 \div 2.5 = 29.64

The quotient is 29.64.

Answer: 29.64

4 · Reviewdoes it hold up?

74.1 and 2.5 add to 76.6 and differ by 71.6, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 5, so 2.5 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 9 hard answer: 3.4

Two numbers have a sum of $79.2$ and a difference of $43.2$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 79.2 and differ by 43.2. We want the bigger one divided by the smaller.

Givens

Their sum is 79.2.
Their difference is 43.2.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

79.2 + 43.2 = 122.4 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

122.4 \div 2 = 61.2

The larger is 61.2.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

79.2 - 61.2 = 18

The smaller is 18.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

61.2 \div 18 = 3.4

The quotient is 3.4.

Answer: 3.4

4 · Reviewdoes it hold up?

61.2 and 18 add to 79.2 and differ by 43.2, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 36, so 18 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 10 hard answer: 42.3

Two numbers have a sum of $86.6$ and a difference of $82.6$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 86.6 and differ by 82.6. We want the bigger one divided by the smaller.

Givens

Their sum is 86.6.
Their difference is 82.6.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

86.6 + 82.6 = 169.2 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

169.2 \div 2 = 84.6

The larger is 84.6.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

86.6 - 84.6 = 2

The smaller is 2.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

84.6 \div 2 = 42.3

The quotient is 42.3.

Answer: 42.3

4 · Reviewdoes it hold up?

84.6 and 2 add to 86.6 and differ by 82.6, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 4, so 2 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 11 hard answer: 43.6

Two numbers have a sum of $89.2$ and a difference of $85.2$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 89.2 and differ by 85.2. We want the bigger one divided by the smaller.

Givens

Their sum is 89.2.
Their difference is 85.2.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

89.2 + 85.2 = 174.4 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

174.4 \div 2 = 87.2

The larger is 87.2.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

89.2 - 87.2 = 2

The smaller is 2.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

87.2 \div 2 = 43.6

The quotient is 43.6.

Answer: 43.6

4 · Reviewdoes it hold up?

87.2 and 2 add to 89.2 and differ by 85.2, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 4, so 2 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.

Variant 12 hard answer: 8.02

Two numbers have a sum of $90.2$ and a difference of $70.2$ . Find the quotient when the larger is divided by the smaller.

Show solution

1 · Understandwhat's really being asked

Two numbers add to 90.2 and differ by 70.2. We want the bigger one divided by the smaller.

Givens

Their sum is 90.2.
Their difference is 70.2.

Unknowns

The quotient of the larger by the smaller.

Constraints

The difference is the larger minus the smaller, so it is positive.

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#11 Work Backwards

Two unknowns and two facts. Adding the facts makes the smaller number cancel -- it appears once with a plus and once with a minus -- so one addition leaves only the larger.

3 · Execute4 carry out the plan

1Add the sum and the difference

#13 Convert to Algebra 6.EE.B.6

The smaller number is added once and subtracted once, so it disappears and twice the larger is left.

90.2 + 70.2 = 160.4 = 2 \times \text{larger}

One unknown gone in one step.

2Halve it

#13 Convert to Algebra 6.EE.B.7

That gives the larger number.

160.4 \div 2 = 80.2

The larger is 80.2.

3Get the smaller from the sum

#7 Identify Subproblems 6.NS.B.3

Take the larger off the total.

90.2 - 80.2 = 10

The smaller is 10.

4Divide

#11 Work Backwards 6.NS.B.3

Now the question can be answered directly.

80.2 \div 10 = 8.02

The quotient is 8.02.

Answer: 8.02

4 · Reviewdoes it hold up?

80.2 and 10 add to 90.2 and differ by 70.2, so both facts hold.

Another way: Subtracting the difference from the sum instead cancels the larger and gives twice the smaller -- 20, so 10 -- the same pair from the other side.

Standardsmin grade 6

6.EE.B.7 Solve one-step equations x+p=q and px=q — Solving for each number once one of them is isolated.
6.EE.B.6 Use variables to represent numbers and write expressions — Treating the two numbers as unknowns and combining the facts.
6.NS.B.3 Fluently add, subtract, multiply, divide multi-digit decimals — Dividing one decimal by another.

💡Takeaway. Given a sum and a difference, add them: the smaller number cancels and the bigger one is left standing.