← Reduce to one unknown to solve · Find Two Unknowns from Sum and Difference

Reduce to one unknown to solve · 10 practice problems

3.OA.D.83.NBT.A.23.OA.A.4

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Generated variants — 10

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

Variant 1 easy

$B$ is $40$ greater than $A$ . When the sum of $A$ and $B$ is $460$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 40 more than A, and together they add to 460. I need to find each number.

Givens

B is 40 greater than A.
A plus B equals 460.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 40.
A + B = 460, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 40 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 40 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 40 from the total

#11 Work Backwards 3.NBT.A.2

B carries 40 more than A. If I subtract that 40 from the total 460, what remains would be two equal copies of A.

460 - 40 = 420

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 420 is two equal amounts of A, so halve it to find A.

420 \div 2 = 210

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 40 back to get B

#11 Work Backwards 3.NBT.A.2

B is 40 more than A, so add 40 to 210.

210 + 40 = 250

Putting the difference back is one more three-digit addition.

Answer: A = 210, B = 250

4 · Reviewdoes it hold up?

Check both conditions: 210 + 250 = 460 (matches the total) and 250 - 210 = 40 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 208, then B = 248 and the sum is 456, short of 460; nudging A up to 210 makes B = 250 and the sum 460, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 460 - 40 and adding 210 + 40 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 420 to find A, the unknown that satisfies 2 x A = 420.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Variant 2 easy

$B$ is $50$ greater than $A$ . When the sum of $A$ and $B$ is $500$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 50 more than A, and together they add to 500. I need to find each number.

Givens

B is 50 greater than A.
A plus B equals 500.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 50.
A + B = 500, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 50 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 50 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 50 from the total

#11 Work Backwards 3.NBT.A.2

B carries 50 more than A. If I subtract that 50 from the total 500, what remains would be two equal copies of A.

500 - 50 = 450

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 450 is two equal amounts of A, so halve it to find A.

450 \div 2 = 225

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 50 back to get B

#11 Work Backwards 3.NBT.A.2

B is 50 more than A, so add 50 to 225.

225 + 50 = 275

Putting the difference back is one more three-digit addition.

Answer: A = 225, B = 275

4 · Reviewdoes it hold up?

Check both conditions: 225 + 275 = 500 (matches the total) and 275 - 225 = 50 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 223, then B = 273 and the sum is 496, short of 500; nudging A up to 225 makes B = 275 and the sum 500, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 500 - 50 and adding 225 + 50 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 450 to find A, the unknown that satisfies 2 x A = 450.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Variant 3 easy

$B$ is $100$ greater than $A$ . When the sum of $A$ and $B$ is $600$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 100 more than A, and together they add to 600. I need to find each number.

Givens

B is 100 greater than A.
A plus B equals 600.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 100.
A + B = 600, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 100 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 100 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 100 from the total

#11 Work Backwards 3.NBT.A.2

B carries 100 more than A. If I subtract that 100 from the total 600, what remains would be two equal copies of A.

600 - 100 = 500

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 500 is two equal amounts of A, so halve it to find A.

500 \div 2 = 250

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 100 back to get B

#11 Work Backwards 3.NBT.A.2

B is 100 more than A, so add 100 to 250.

250 + 100 = 350

Putting the difference back is one more three-digit addition.

Answer: A = 250, B = 350

4 · Reviewdoes it hold up?

Check both conditions: 250 + 350 = 600 (matches the total) and 350 - 250 = 100 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 248, then B = 348 and the sum is 596, short of 600; nudging A up to 250 makes B = 350 and the sum 600, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 600 - 100 and adding 250 + 100 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 500 to find A, the unknown that satisfies 2 x A = 500.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Variant 4 medium

$B$ is $88$ greater than $A$ . When the sum of $A$ and $B$ is $700$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 88 more than A, and together they add to 700. I need to find each number.

Givens

B is 88 greater than A.
A plus B equals 700.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 88.
A + B = 700, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 88 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 88 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 88 from the total

#11 Work Backwards 3.NBT.A.2

B carries 88 more than A. If I subtract that 88 from the total 700, what remains would be two equal copies of A.

700 - 88 = 612

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 612 is two equal amounts of A, so halve it to find A.

612 \div 2 = 306

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 88 back to get B

#11 Work Backwards 3.NBT.A.2

B is 88 more than A, so add 88 to 306.

306 + 88 = 394

Putting the difference back is one more three-digit addition.

Answer: A = 306, B = 394

4 · Reviewdoes it hold up?

Check both conditions: 306 + 394 = 700 (matches the total) and 394 - 306 = 88 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 304, then B = 392 and the sum is 696, short of 700; nudging A up to 306 makes B = 394 and the sum 700, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 700 - 88 and adding 306 + 88 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 612 to find A, the unknown that satisfies 2 x A = 612.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Variant 5 medium

$B$ is $124$ greater than $A$ . When the sum of $A$ and $B$ is $800$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 124 more than A, and together they add to 800. I need to find each number.

Givens

B is 124 greater than A.
A plus B equals 800.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 124.
A + B = 800, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 124 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 124 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 124 from the total

#11 Work Backwards 3.NBT.A.2

B carries 124 more than A. If I subtract that 124 from the total 800, what remains would be two equal copies of A.

800 - 124 = 676

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 676 is two equal amounts of A, so halve it to find A.

676 \div 2 = 338

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 124 back to get B

#11 Work Backwards 3.NBT.A.2

B is 124 more than A, so add 124 to 338.

338 + 124 = 462

Putting the difference back is one more three-digit addition.

Answer: A = 338, B = 462

4 · Reviewdoes it hold up?

Check both conditions: 338 + 462 = 800 (matches the total) and 462 - 338 = 124 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 336, then B = 460 and the sum is 796, short of 800; nudging A up to 338 makes B = 462 and the sum 800, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 800 - 124 and adding 338 + 124 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 676 to find A, the unknown that satisfies 2 x A = 676.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Variant 6 medium

$B$ is $197$ greater than $A$ . When the sum of $A$ and $B$ is $801$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 197 more than A, and together they add to 801. I need to find each number.

Givens

B is 197 greater than A.
A plus B equals 801.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 197.
A + B = 801, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 197 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 197 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 197 from the total

#11 Work Backwards 3.NBT.A.2

B carries 197 more than A. If I subtract that 197 from the total 801, what remains would be two equal copies of A.

801 - 197 = 604

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 604 is two equal amounts of A, so halve it to find A.

604 \div 2 = 302

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 197 back to get B

#11 Work Backwards 3.NBT.A.2

B is 197 more than A, so add 197 to 302.

302 + 197 = 499

Putting the difference back is one more three-digit addition.

Answer: A = 302, B = 499

4 · Reviewdoes it hold up?

Check both conditions: 302 + 499 = 801 (matches the total) and 499 - 302 = 197 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 300, then B = 497 and the sum is 797, short of 801; nudging A up to 302 makes B = 499 and the sum 801, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 801 - 197 and adding 302 + 197 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 604 to find A, the unknown that satisfies 2 x A = 604.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Variant 7 medium

$B$ is $66$ greater than $A$ . When the sum of $A$ and $B$ is $888$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 66 more than A, and together they add to 888. I need to find each number.

Givens

B is 66 greater than A.
A plus B equals 888.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 66.
A + B = 888, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 66 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 66 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 66 from the total

#11 Work Backwards 3.NBT.A.2

B carries 66 more than A. If I subtract that 66 from the total 888, what remains would be two equal copies of A.

888 - 66 = 822

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 822 is two equal amounts of A, so halve it to find A.

822 \div 2 = 411

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 66 back to get B

#11 Work Backwards 3.NBT.A.2

B is 66 more than A, so add 66 to 411.

411 + 66 = 477

Putting the difference back is one more three-digit addition.

Answer: A = 411, B = 477

4 · Reviewdoes it hold up?

Check both conditions: 411 + 477 = 888 (matches the total) and 477 - 411 = 66 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 409, then B = 475 and the sum is 884, short of 888; nudging A up to 411 makes B = 477 and the sum 888, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 888 - 66 and adding 411 + 66 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 822 to find A, the unknown that satisfies 2 x A = 822.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Variant 8 hard

$B$ is $156$ greater than $A$ . When the sum of $A$ and $B$ is $902$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 156 more than A, and together they add to 902. I need to find each number.

Givens

B is 156 greater than A.
A plus B equals 902.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 156.
A + B = 902, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 156 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 156 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 156 from the total

#11 Work Backwards 3.NBT.A.2

B carries 156 more than A. If I subtract that 156 from the total 902, what remains would be two equal copies of A.

902 - 156 = 746

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 746 is two equal amounts of A, so halve it to find A.

746 \div 2 = 373

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 156 back to get B

#11 Work Backwards 3.NBT.A.2

B is 156 more than A, so add 156 to 373.

373 + 156 = 529

Putting the difference back is one more three-digit addition.

Answer: A = 373, B = 529

4 · Reviewdoes it hold up?

Check both conditions: 373 + 529 = 902 (matches the total) and 529 - 373 = 156 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 371, then B = 527 and the sum is 898, short of 902; nudging A up to 373 makes B = 529 and the sum 902, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 902 - 156 and adding 373 + 156 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 746 to find A, the unknown that satisfies 2 x A = 746.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Variant 9 hard

$B$ is $210$ greater than $A$ . When the sum of $A$ and $B$ is $950$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 210 more than A, and together they add to 950. I need to find each number.

Givens

B is 210 greater than A.
A plus B equals 950.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 210.
A + B = 950, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 210 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 210 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 210 from the total

#11 Work Backwards 3.NBT.A.2

B carries 210 more than A. If I subtract that 210 from the total 950, what remains would be two equal copies of A.

950 - 210 = 740

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 740 is two equal amounts of A, so halve it to find A.

740 \div 2 = 370

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 210 back to get B

#11 Work Backwards 3.NBT.A.2

B is 210 more than A, so add 210 to 370.

370 + 210 = 580

Putting the difference back is one more three-digit addition.

Answer: A = 370, B = 580

4 · Reviewdoes it hold up?

Check both conditions: 370 + 580 = 950 (matches the total) and 580 - 370 = 210 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 368, then B = 578 and the sum is 946, short of 950; nudging A up to 370 makes B = 580 and the sum 950, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 950 - 210 and adding 370 + 210 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 740 to find A, the unknown that satisfies 2 x A = 740.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Variant 10 hard

$B$ is $300$ greater than $A$ . When the sum of $A$ and $B$ is $1000$ , find $A$ and $B$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Two numbers A and B: B is 300 more than A, and together they add to 1000. I need to find each number.

Givens

B is 300 greater than A.
A plus B equals 1000.

Unknowns

The value of A.
The value of B.

Constraints

B = A + 300.
A + B = 1000, with A and B whole numbers.

2 · Planchoose the strategy

#11 Work Backwards · also uses: #6 Guess and Check

If I imagine taking the extra 300 off of B first, the two numbers would be equal, so the leftover total splits evenly in half to give A. Then add 300 back to get B. Working backwards from the total turns the two-unknown problem into one tidy halving plus an addition.

3 · Execute3 carry out the plan

1Remove the extra 300 from the total

#11 Work Backwards 3.NBT.A.2

B carries 300 more than A. If I subtract that 300 from the total 1000, what remains would be two equal copies of A.

1000 - 300 = 700

Peeling off the known difference is a single Grade 3 subtraction within 1000.

2Split the rest into two equal parts to get A

#11 Work Backwards 3.OA.A.4

The 700 is two equal amounts of A, so halve it to find A.

700 \div 2 = 350

Sharing a number into 2 equal groups is basic Grade 3 division thinking.

3Add the 300 back to get B

#11 Work Backwards 3.NBT.A.2

B is 300 more than A, so add 300 to 350.

350 + 300 = 650

Putting the difference back is one more three-digit addition.

Answer: A = 350, B = 650

4 · Reviewdoes it hold up?

Check both conditions: 350 + 650 = 1000 (matches the total) and 650 - 350 = 300 (matches the difference). Both hold, so the pair is correct.

Another way: Guess and check (Tool 6): try A = 348, then B = 648 and the sum is 996, short of 1000; nudging A up to 350 makes B = 650 and the sum 1000, confirming the answer.

Standardsmin grade 3

3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 1000 - 300 and adding 350 + 300 with three-digit numbers.
3.OA.A.4 Determine unknown whole number in multiplication or division equation — Halving 700 to find A, the unknown that satisfies 2 x A = 700.

💡Takeaway. Take the gap off the total, share what's left in half, then add the gap back -- pure Grade 3 thinking!

Where next?

← Back to the problem: Reduce to one unknown to solve