4.4 Ratio Reasoning with Tables, Tape Diagrams, & Double Number Lines

Ratio Reasoning with Tables, Tape Diagrams, & Double Number Lines

Section 4.1 solved proportions algebraically. That is one of four representations the DRC specification lists for Level D standard 6.RP.3, which asks candidates to "use ratio and rate reasoning to solve real-world and mathematical problems, e.g., by reasoning about tables of equivalent ratios, tape diagrams, double number line diagrams, or equations." TABE items are written around these models, so recognizing them is worth points even when you would rather cross-multiply.

Representation 1: Tables of Equivalent Ratios

A ratio table lists pairs that all reduce to the same ratio. Scaling any column by the same factor produces another valid row.

Concrete mix at 1 part cement to 4 parts aggregate:

Reading the table sideways answers scaling questions. Reading it downward exposes the constant multiplier: every aggregate entry is 4 times its cement entry, so $a = 4c$.

Ratio tables also let you combine columns. Need 7 bags of cement? Add the "5" column and the "2" column: $20 + 8 = 28$ bags of aggregate. This additive shortcut is legitimate because equivalent ratios are proportional.

Representation 2: Tape Diagrams

A tape diagram (also called a bar model or strip diagram) draws each part of a ratio as a run of identical boxes. It turns a sharing problem into simple division.

Problem. A $96 bonus is divided between two technicians in the ratio 3 : 5. How much does each receive?

Technician A:  [ 12 ][ 12 ][ 12 ]                     = 36
Technician B:  [ 12 ][ 12 ][ 12 ][ 12 ][ 12 ]         = 60
               |--------- 8 units = $96 ---------|

Method:

1. Count total units: $3 + 5 = 8$ units.
2. Find one unit: $96 \div 8 = 12$ dollars per unit.
3. Scale each share: A gets $3 \times 12 = $36$; B gets $5 \times 12 = $60$.
4. Check: $36 + 60 = 96$. ✓

The same three steps handle every "divide in the ratio" item on TABE, including three-way splits: a 2 : 3 : 7 division of 144 uses 12 units of 12.

Tape diagrams for difference problems

A parking lot holds cars and trucks in a 7 : 4 ratio. There are 63 more cars than trucks. How many vehicles in total?

The difference is $7 - 4 = 3$ units, and that difference is 63 vehicles, so one unit is $63 \div 3 = 21$. Total units: $7 + 4 = 11$, giving $11 \times 21 = \mathbf{231}$ vehicles.

Notice you never needed the individual counts. Matching the given number to the right number of units is the whole skill.

Representation 3: Double Number Lines

A double number line stacks two parallel scales with matching tick marks, one for each unit. It is the natural model for rates.

Miles:   0 ────── 45 ────── 90 ────── 135 ────── 180
Hours:   0 ─────── 1 ─────── 2 ─────── 3 ─────── 4

• The unit rate is whatever sits above 1: 45 miles per hour.
• To find distance in 2.5 hours, land between ticks: $45 \times 2.5 = 112.5$ miles.
• To find time for 315 miles, work the other direction: $315 \div 45 = 7$ hours.

Double number lines are especially good at percent items, because percent is a rate out of 100:

Dollars:  0 ─────── 18 ─────── 36 ─────── 45 ─────── 90
Percent:  0% ────── 20% ────── 40% ────── 50% ───── 100%

Reading off that 45 aligns with 50% tells you the whole is 90 without writing a single equation.

Representation 4: The Equation

Every ratio table has an equation hiding in it. If $y$ is always $k$ times $x$, then $y = kx$, and $k$ is the constant of proportionality — the unit rate.

Choosing the Right Tool Under Time Pressure

• The problem says "in the ratio a : b" and gives a total or a difference → tape diagram.
• The problem gives a rate and asks for a scaled value → double number line or $y = kx$.
• The problem gives a table and asks whether it is proportional → check that $y \div x$ is constant in every row.
• The problem gives two complete pairs and one unknown → cross-multiply.