Reduce to one unknown to solve

3.OA.D.83.NBT.A.23.OA.A.4 · take · grade 3

Archetype: Find Two Unknowns from Sum and Difference · step in a 8-type progression

▶ Practice — 10 problems

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

Show solution

Understand

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.

Plan

#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.

Execute

#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.

#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.

#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

Review

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

Guess and check (Tool 6): try A = 300, then B = 497 and the sum is 797, which is 4 short of 801; nudging A up by 2 to 302 makes B = 499 and the sum 801, confirming the answer.

Standards · min 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.

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