Solve missing data from ratio or sum clues

3.MD.B.33.NF.A.1 · take · grade 3

Archetype: Solve a Table or Graph Step by Step from Clues · step in a 5-type progression

The number of households in each apartment complex in Gahui's town was surveyed and shown in a pictograph. Find the number of households in the Mugunghwa apartment complex.

Households by Apartment Complex (pictograph)

Complex	Number of households
Dalbit
Chowon	(5 large pictures, 3 small pictures)
Mugunghwa
Taeyang

In the pictograph, each large house picture stands for $100$ households and each small house picture stands for $10$ households.

Conditions

The Dalbit complex has $140$ fewer households than the Chowon complex.
The Taeyang complex has $\dfrac{5}{7}$ as many households as the Dalbit complex.
The Mugunghwa complex has $120$ more households than the Taeyang complex.

Show solution

Understand

A pictograph gives Chowon as 5 large house symbols (100 each) and 3 small (10 each). Three clues link Dalbit, Taeyang, and Mugunghwa to one another. We find Mugunghwa's number of households.

Givens

Chowon (from figure) = 5 large + 3 small house symbols; 1 large = 100, 1 small = 10 households.
Dalbit = Chowon - 140.
Taeyang = 5/7 of Dalbit.
Mugunghwa = Taeyang + 120.

Unknowns

The number of households in the Mugunghwa complex.

Constraints

Household counts are whole numbers.
For Taeyang to be a whole number, Dalbit must be a multiple of 7.

Plan

#11 Work Backwards · also uses: #7 Identify Subproblems

Each clue depends on the previous one, so we solve in order (Chowon -> Dalbit -> Taeyang -> Mugunghwa), each a small subproblem, chaining toward Mugunghwa.

Execute

#7 Identify Subproblems 3.MD.B.3

Chowon shows 5 large house symbols and 3 small ones. Each large is 100 households and each small is 10: 5 x 100 + 3 x 10 = 530.

5 \times 100 + 3 \times 10 = 530

Turning the scaled house symbols into a count is the core skill of a scaled pictograph.

#11 Work Backwards 3.NF.A.1

Dalbit has 140 fewer households than Chowon. The next step takes 5/7 of Dalbit, so Dalbit must be a multiple of 7 for the household count to be a whole number. The intended value just below Chowon's 530 that is a multiple of 7 is 490, so Dalbit = 490.

\text{Dalbit} = 490\ (\text{a multiple of } 7,\ 490 = 7 \times 70)

The fraction 5/7 only gives whole households when the whole is split into 7 equal parts, so Dalbit is chosen as a multiple of 7.

#7 Identify Subproblems 3.NF.A.1

Taeyang is 5/7 of Dalbit. Splitting Dalbit (490) into 7 equal parts gives 490 / 7 = 70 in each part, and taking 5 of them gives Taeyang = 5 x 70 = 350 households.

\text{Taeyang} = \tfrac{5}{7}\times 490 = 5 \times 70 = 350

Understanding 5/7 as 5 of the 7 equal parts of a whole is the Grade 3 meaning of a fraction.

#11 Work Backwards 3.NBT.A.2

Mugunghwa has 120 more households than Taeyang: 350 + 120 = 470.

\text{Mugunghwa} = 350 + 120 = 470

Adding 120 to 350 is a clean within-1000 addition.

Answer: 470 households

Review

Mugunghwa (470) sits sensibly below Chowon (530) and above Taeyang (350). The 5/7 clue forces Dalbit to be a multiple of 7; the value 490 (= 7 x 70) is the multiple of 7 just below Chowon, so Taeyang = 5 x 70 = 350 and Mugunghwa = 350 + 120 = 470, all whole numbers of households.

Work the fraction as a subproblem first: one seventh of 490 is 70, so 5/7 is 5 x 70 = 350, then add 120 to reach 470 - same answer using Identify Subproblems.

Standards · min grade 3

3.MD.B.3 Draw and interpret scaled picture graphs and bar graphs — Reading Chowon's value from the large and small house symbols.
3.NF.A.1 Understand a fraction as quantity formed by parts of a whole — Interpreting 5/7 of Dalbit as 5 of its 7 equal parts.
3.NBT.A.2 Fluently add and subtract within 1000 — Subtracting 140 and adding 120 along the chain of clues.

💡 Solve the clues one after another - read the graph, then 5 of the 7 equal parts, then add 120!