Set up one-unknown equation for capacities

3.MD.A.23.OA.D.8 · adapt · grade 3

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

A total of $64$ fluid ounces of cooking oil is poured into two bottles whose capacities differ by $8$ fluid ounces. If both bottles are filled to the top, how many fluid ounces of cooking oil are in the smaller bottle?

Show solution

Understand

64 fl oz of cooking oil is split between two bottles whose capacities differ by 8 fl oz, and both are filled to the top. Find how many fluid ounces are in the smaller bottle.

Givens

The two bottles together hold 64 fl oz when both are full.
The two capacities differ by 8 fl oz.
Both bottles are filled to the top.

Unknowns

The capacity (oil amount) of the smaller bottle in fluid ounces.

Constraints

The larger bottle holds exactly 8 fl oz more than the smaller one.
The two amounts add up to 64 fl oz.

Plan

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

If we set aside the extra 8 fl oz of the bigger bottle, the two bottles would be equal. Remove that 8 fl oz from the total, split what is left in half, and that half is the smaller bottle.

Execute

#11 Work Backwards 3.OA.D.8

The larger bottle has 8 fl oz more than the smaller. Take those 8 fl oz away from the total of 64 fl oz, leaving 64 - 8 = 56 fl oz to share as two equal smaller bottles.

64 - 8 = 56\ \text{fl oz}

Setting aside the extra makes both bottles the same size, which is easier to share.

#11 Work Backwards 3.OA.A.3

The 56 fl oz now stands for two equal smaller bottles, so each smaller bottle holds 56 divided by 2 = 28 fl oz.

56 \div 2 = 28\ \text{fl oz}

Two equal parts means cut the leftover total in half.

Answer: 28 fl oz

Review

Smaller bottle 28 fl oz and larger bottle 28 + 8 = 36 fl oz add to 28 + 36 = 64 fl oz, and they differ by 8 fl oz, so 28 fl oz checks out.

Guess and check (tool 6): try smaller = 28, larger = 36, sum 64, difference 8; both conditions fit, confirming 28 fl oz.

Standards · min grade 3

3.OA.D.8 Solve two-step word problems using four operations within 100 — Subtracting the 8 fl oz difference then halving to find the smaller bottle.
3.OA.A.3 Solve multiplication and division word problems within 100 — Dividing the equalized total by 2.

💡 This only needs Grade 3 sense: take away the extra, then split the rest in half!