Representing Ratios - Pre Algebra

1. Which operation on both numbers in a ratio table row always produces an equivalent ratio?

   Multiplying or dividing both numbers by the same nonzero number. Adding the same number to both does not keep the ratio.

2. On a graph of a ratio relationship, what does the point $(1, y)$ represent?

   The unit rate: the amount of the second quantity for exactly 1 unit of the first.

3. When the pairs in a table of equivalent ratios are graphed as points, what pattern do they form?

   They lie on a straight line through the origin, because every pair has the same ratio.

4. A ratio table pairs 3 cups of flour with 2 cups of milk. How many cups of milk go with 12 cups of flour?

   8 cups. Flour is multiplied by 4, so milk is $2 \times 4 = 8$.

5. A ratio table shows 3 tickets cost 24 dollars. What do 7 tickets cost at the same rate?

   56 dollars. One ticket costs $24 \div 3 = 8$ dollars, and $7 \times 8 = 56$.

6. Why do the pairs $(2, 3)$ and $(4, 7)$ not belong in the same ratio table?

   Doubling $2:3$ gives $4:6$, not $4:7$. Their ratios $\frac{3}{2} = 1.5$ and $\frac{7}{4} = 1.75$ differ.

7. The points $(2, 5)$ and $(6, y)$ represent the same ratio. What is $y$?

   $15$. The first coordinate is multiplied by 3, so $5 \times 3 = 15$.

8. What does a double number line show about a ratio?

   Two quantities on parallel number lines, with equivalent ratios lined up directly above and below each other.

9. The ratio of $x$ to $y$ is $3:4$. Which point represents it with $x$ on the horizontal axis?

   $(3, 4)$. Order matters: $(4, 3)$ would represent the ratio $4:3$.

10. A runner completes 4 laps in 6 minutes. With laps on the $x$-axis and minutes on the $y$-axis, which point shows 2 laps at the same rate?

    $(2, 3)$. Halving both quantities keeps the ratio: 2 laps in 3 minutes.

11. Two quantities are in a $2:5$ ratio. What is the second quantity when the first is 5?

    $12.5$. Multiply both terms by $2.5$: $5 \times 2.5 = 12.5$.

12. A ratio table has the rows $(4, 10)$ and $(6, 15)$. What second value goes with a first value of 10?

    $25$. Adding the two rows gives $(10, 25)$, and $\frac{25}{10} = 2.5$ matches the ratio.

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