Proportionality
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Texas 6th Grade Math › Proportionality
Pattern A:
- Verbal: A $2$-dollar fee is added to the cost, so you pay $x$ dollars plus $2$.
- Table: $x$: 1, 2, 5; $y$: 3, 4, 7
- Equation: $y = x + 2$
Pattern B:
- Verbal: You earn $12$ dollars for each hour worked.
- Table: $x$: 1, 2, 5; $y$: 12, 24, 60
- Equation: $y = 12x$
Which pattern shows an additive relationship?
Pattern B only
Both patterns
Neither patterns
Pattern A only
Explanation
Additive relationships have the form $y = x + a$ (constant $y - x$). Pattern A has $y = x + 2$ so $y - x = 2$. Pattern B is $y = 12x$ (multiplicative) with constant ratio $y/x = 12$ for $x>0$.
A printer prints 45 pages in 3 minutes. What is the unit rate in pages per minute?
15 pages/minute
3 minutes/page
48 pages/minute
12 pages/minute
Explanation
A rate compares different attributes by division. Compute the unit rate by dividing pages by minutes: $45 \div 3 = 15$. As a quotient, that is $\frac{45}{3}$ pages per minute, so 15 pages/minute.
Sarah has 12 stickers and Jake has 4 stickers. What is the ratio of Sarah's stickers to Jake's stickers?
12 to 4
8 more stickers
4:12
12-4
Explanation
Ratios compare quantities multiplicatively. Sarah to Jake is 12 to 4, which can also be written 12:4 or 12/4 and simplified to 3:1, meaning Sarah has 3 times as many stickers.
In a jar, 15% of the marbles are blue. There are 12 blue marbles. What is the total number of marbles?
80
60
72
90
Explanation
Identify part, whole, percent: part = 12, percent = 15%, whole = x. Equation method: $0.15x = 12 \Rightarrow x = 12 \div 0.15 = 80$. Proportion method: $\frac{15}{100} = \frac{12}{x}$, so $15x = 1200$ and $x = 80$. Unit-percent reasoning: if 15% is 12, then 1% is $12 \div 15 = 0.8$, so 100% is $0.8 \times 100 = 80$.
On a map, 1 inch represents 25 miles. The distance between two cities on the map is 3.5 inches. Let $m$ be the actual distance in miles.
Which proportion represents this situation?
$\frac{1}{25} = \frac{3.5}{m}$
$\frac{1}{25} = \frac{m}{3.5}$
$\frac{25}{1} = \frac{3.5}{m}$
$\frac{m}{25} = \frac{1}{3.5}$
Explanation
The ratio inches/miles must stay consistent: $\frac{1}{25} = \frac{3.5}{m}$. Cross-multiplying gives $m = 3.5 \times 25 = 87.5$. The same relationship can be shown with a scale factor (multiply inches by 25 to get miles), a table (Inches: 1, 2, 3.5; Miles: 25, 50, 87.5), or a graph of a line through the origin with slope 25 miles per inch.
A $50 backpack is discounted by 30%. Which fraction represents this discount rate?
$\frac{3}{100}$
$\frac{3}{10}$
$\frac{30}{3}$
$\frac{1}{3}$
Explanation
Percent to fraction: $30% = \frac{30}{100} = \frac{3}{10}$. $\frac{3}{100}$ is 3%, and $\frac{1}{3}$ is about 33.3%.
In a class of 20 students, 8 play soccer. What percent of students play soccer?
$35%$
$60%$
$40%$
$8%$
Explanation
Part-to-whole is $\frac{8}{20}$. Simplify: $\frac{8}{20}=\frac{2}{5}$. Decimal: $2\div 5=0.4$. Percent: $0.4\times 100=40%$. So $40%$ is correct. $60%$ mixes in the non-playing group, and $8%$ confuses the count with a percent.
Four friends equally share one pizza.
What percent of the pizza does each friend get?
20%
25%
30%
50%
Explanation
Each friend gets 1/4 of the pizza. 1 ÷ 4 = 0.25, and 0.25 × 100 = 25%, so 25% is equivalent to 1/4 and 0.25.
How many inches are in 3.5 feet? Use 12 inches = 1 foot.
42 inches
35 inches
36 inches
0.29 inches
Explanation
Unit rate: $3.5\text{ ft}\times \frac{12\text{ in}}{1\text{ ft}}=42\text{ in}$ (ft cancels). Proportion: $\frac{3.5\text{ ft}}{x\text{ in}}=\frac{1\text{ ft}}{12\text{ in}}\Rightarrow x=3.5\times 12=42$.
A car travels 240 miles in 4 hours. At this rate, how far will it travel in 7 hours? Using proportional reasoning, what is the answer?
420 miles
360 miles
300 miles
480 miles
Explanation
Find the unit rate: $240 \div 4 = 60$ miles/hour. Predict for 7 hours: $60 \times 7 = 420$ miles. Proportion: $\frac{240}{4} = \frac{x}{7} \Rightarrow x = \frac{240\cdot 7}{4} = 420$.