if two quantities are proportional, then their relationship can be represented by an equation of the form $y = kcdot x$
(a) assume that the number of miles driven is proportional to the gas used by the car. if a car can drive 66 miles on 3 gallons of gas, find an equation that represents this relationship
(b) assume that the water used in a pancake recipe is proportional to the flour used at a ratio of 6 cups of water to 7 cups of flour. find an equation that represents this relationship
(c) assume that the ratio of currency a to currency b to currency c is 5 : 6 : 900 and that these currencies are proportional to each other. find equations that represent relationships between pairs of these quantities.

(b) choose the correct answer below
a. $w = 7cdot f$
b. $w = 6cdot f$
c. $w=\frac{6}{7}cdot f$
d. $w=\frac{7}{6}cdot f$

(c) select all that apply
a. $a = 6cdot b$
b. $c = 150cdot a$
c. $a = 180cdot c$
d. $b=\frac{6}{5}a$
e. $b = 150cdot c$
f. $b=\frac{1}{150}cdot c$
g. $c = 150cdot b$

Question

if two quantities are proportional, then their relationship can be represented by an equation of the form $y = kcdot x$
(a) assume that the number of miles driven is proportional to the gas used by the car. if a car can drive 66 miles on 3 gallons of gas, find an equation that represents this relationship
(b) assume that the water used in a pancake recipe is proportional to the flour used at a ratio of 6 cups of water to 7 cups of flour. find an equation that represents this relationship
(c) assume that the ratio of currency a to currency b to currency c is 5 : 6 : 900 and that these currencies are proportional to each other. find equations that represent relationships between pairs of these quantities.

(b) choose the correct answer below
a. $w = 7cdot f$
b. $w = 6cdot f$
c. $w=\frac{6}{7}cdot f$
d. $w=\frac{7}{6}cdot f$

(c) select all that apply
a. $a = 6cdot b$
b. $c = 150cdot a$
c. $a = 180cdot c$
d. $b=\frac{6}{5}a$
e. $b = 150cdot c$
f. $b=\frac{1}{150}cdot c$
g. $c = 150cdot b$

Accepted Answer

### Part (a)
# Explanation:
## Step 1: Identify variables and constant of proportionality
Let $ y $ be the number of miles driven and $ x $ be the gas used (in gallons). The proportional relationship is $ y = kx $. We know when $ x = 3 $, $ y = 66 $. To find $ k $, we substitute these values into the equation: $ 66 = k \times 3 $.
## Step 2: Solve for $ k $
Divide both sides by 3: $ k=\frac{66}{3}=22 $. So the equation is $ y = 22x $ (where $ y $ is miles and $ x $ is gallons of gas).
# Answer: $ y = 22x $ (or using appropriate variable names like $ \text{miles} = 22 \times \text{gas} $)

### Part (b)
# Brief Explanations:
We have water ($ W $) proportional to flour ($ F $) with a ratio of 6 cups water to 7 cups flour. The proportional equation is $ W = kF $, where $ k=\frac{\text{water}}{\text{flour}}=\frac{6}{7} $. So $ W=\frac{6}{7}F $, which matches option C.
# Answer: C. $ W=\frac{6}{7}\cdot F $

### Part (c)
The ratio of $ A:B:C = 5:6:900 $. Let's analyze each option:
- **Option A**: $ A = 6B $? From $ A:B = 5:6 $, we get $ A=\frac{5}{6}B $, so A is wrong.
- **Option B**: $ C = 150A $? From $ A:C = 5:900 $, $ C=\frac{900}{5}A = 180A $? Wait, no: $ \frac{C}{A}=\frac{900}{5}=180 $? Wait, $ 900\div5 = 180 $? Wait, no, $ 900\div5 = 180 $? Wait, 5:900 for A:C, so $ C=\frac{900}{5}A=180A $? Wait, no, let's recalculate. Wait, $ A:B:C = 5:6:900 $, so $ \frac{A}{B}=\frac{5}{6}\implies B=\frac{6}{5}A $ (Option D), $ \frac{A}{C}=\frac{5}{900}=\frac{1}{180}\implies C = 180A $? Wait, no, $ \frac{C}{A}=\frac{900}{5}=180 \implies C = 180A $, so Option B says $ C = 150A $, which is wrong. Wait, maybe I miscalculated the ratio. Wait, $ A:B:C = 5:6:900 $, so $ \frac{B}{A}=\frac{6}{5}\implies B=\frac{6}{5}A $ (Option D is correct). $ \frac{C}{B}=\frac{900}{6}=150\implies C = 150B $ (Option G: $ C = 150B $, correct? Wait, $ B:C = 6:900 = 1:150 $, so $ C = 150B $, so G is correct? Wait, let's re-express the ratios:
  - $ A:B = 5:6 \implies B=\frac{6}{5}A $ (Option D: $ B=\frac{6}{5}A $, correct).
  - $ B:C = 6:900 = 1:150 \implies C = 150B $ (Option G: $ C = 150B $, correct).
  - $ A:C = 5:900 = 1:180 \implies C = 180A $, so Option B ($ C = 150A $) is wrong, Option C ($ A = 180C $) is wrong (it should be $ C = 180A $).
  - Option E: $ B = 150C $? From $ B:C = 6:900 $, $ B=\frac{6}{900}C=\frac{1}{150}C $, so E is wrong, F is $ B=\frac{1}{150}C $, which is correct? Wait, let's redo:

From $ A:B:C = 5:6:900 $:
- $ \frac{A}{B}=\frac{5}{6}\implies B=\frac{6}{5}A $ (Option D: $ B=\frac{6}{5}A $, correct).
- $ \frac{B}{C}=\frac{6}{900}=\frac{1}{150}\implies B=\frac{1}{150}C $ (Option F: $ B=\frac{1}{150}\cdot C $, correct).
- $ \frac{C}{B}=\frac{900}{6}=150\implies C = 150B $ (Option G: $ C = 150\cdot B $, correct).
- $ \frac{C}{A}=\frac{900}{5}=180\implies C = 180A $, so Option B ($ C = 150A $) is wrong, Option A is wrong, Option C is wrong.

Wait, maybe I made a mistake earlier. Let's list all correct:
- Option D: $ B=\frac{6}{5}A $ (from $ A:B = 5:6 \implies B=\frac{6}{5}A $) – correct.
- Option F: $ B=\frac{1}{150}C $ (from $ B:C = 6:900 \implies B=\frac{6}{900}C=\frac{1}{150}C $) – correct.
- Option G: $ C = 150B $ (from $ B:C = 6:900 \implies C=\frac{900}{6}B = 150B $) – correct.

Wait, but let's check again:

1. $ A:B = 5:6 \implies \frac{A}{B}=\frac{5}{6}\implies B=\frac{6}{5}A $ (Option D: correct).
2. $ B:C = 6:900 \implies \frac{B}{C}=\frac{6}{900}=\frac{1}{150}\implies B=\frac{1}{150}C $ (Option F: correct) and $ C = 150B $ (Option G: correct).
3. $ A:C = 5:900 \implies \frac{A}{C}=\frac{5}{900}=\frac{1}{180}\implies C = 180A $, so Option B ($ C = 150A $) is wrong, Option C ($ A = 180C $) is wrong (it's $ C = 180A $).

So correct options are D, F, G? Wait, no, let's check the original ratio again. Wait, the ratio is $ A:B:C = 5:6:900 $, so:

- $ A = 5k $, $ B = 6k $, $ C = 900k $ for some $ k $.

Now check each option:

- **Option D**: $ B=\frac{6}{5}A $. Substitute $ A = 5k $, $ B = 6k $: $ 6k=\frac{6}{5}(5k)=6k $ – correct.
- **Option F**: $ B=\frac{1}{150}C $. Substitute $ C = 900k $: $ \frac{1}{150}(900k)=6k = B $ – correct.
- **Option G**: $ C = 150B $. Substitute $ B = 6k $: $ 150(6k)=900k = C $ – correct.

Other options:

- A: $ A = 6B \implies 5k = 6(6k)=36k $ – false.
- B: $ C = 150A \implies 900k = 150(5k)=750k $ – false.
- C: $ A = 180C \implies 5k = 180(900k) $ – false.
- E: $ B = 150C \implies 6k = 150(900k) $ – false.

So correct options are D, F, G.

# Brief Explanations:
Using the ratio $ A:B:C = 5:6:900 $, we express relationships as $ A = 5k $, $ B = 6k $, $ C = 900k $. Checking each option:
- D: $ B=\frac{6}{5}A $ (substitute $ A = 5k $, $ B = 6k $, holds).
- F: $ B=\frac{1}{150}C $ (substitute $ C = 900k $, $ \frac{1}{150}(900k)=6k = B $, holds).
- G: $ C = 150B $ (substitute $ B = 6k $, $ 150(6k)=900k = C $, holds).
# Answer: D. $ B=\frac{6}{5}A $, F. $ B=\frac{1}{150}\cdot C $, G. $ C = 150\cdot B $

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QUESTION IMAGE

Question

Explanation:

Part (a)

Step 1: Identify variables and constant of proportionality

Step 2: Solve for \( k \)

Answer:

Part (b)