Question

problem 9 part a: spiral review
for each description, decide whether it matches diagram a, diagram b, or neither.
x represents the original value and y represents the final (new) value.
an increase by 25%
diagram a diagram b neither
a decrease by 25%
diagram a diagram b neither
y = 1.\overline{6}x
diagram a diagram b neither

Explanation:

1. An increase by 25%

• Analyze Diagram A: \( x \) has 4 parts, \( y \) has 3 parts. So \( y=\frac{3}{4}x = 0.75x \) (decrease by 25%).
• Analyze Diagram B: \( x \) has 4 parts, \( y \) has 5 parts. So \( y=\frac{5}{4}x = 1.25x \) (increase by 25%? Wait, 25% increase is \( 1.25x \), but let's check the second part. Wait, no: 25% increase: original is \( x \), new is \( x + 0.25x=1.25x \). Diagram B: \( x \) is 4 units, \( y \) is 5 units? Wait, no, Diagram B's \( x \) is 4 red parts, and \( y \) is 4 red +1 white? Wait, no, the diagram: Diagram A: \( x \) is 4 red rectangles, \( y \) is 3 red rectangles. So \( y=\frac{3}{4}x \) (decrease by 25%). Diagram B: \( x \) is 4 red rectangles, \( y \) is 5 rectangles (4 red +1 white). So \( y=\frac{5}{4}x = 1.25x \) (increase by 25%? Wait, 25% increase is 1.25x. But wait, the first question: "An increase by 25%". So which diagram? Diagram B: \( y = \frac{5}{4}x = 1.25x \), which is a 25% increase (since \( 1.25x = x + 0.25x \)). Wait, no: 25% increase: original \( x \), new \( y = x + 0.25x = 1.25x \). Diagram B: \( x \) is 4 parts, \( y \) is 5 parts. So \( y = \frac{5}{4}x = 1.25x \), so that's a 25% increase. Wait, but wait, the first description: "An increase by 25%". So Diagram B? Wait, no, let's re-examine:

Wait, Diagram A: \( x \) (original) is 4 units, \( y \) (new) is 3 units. So \( y = \frac{3}{4}x = 0.75x \), which is a 25% decrease (since \( 1 - 0.75 = 0.25 \)). Diagram B: \( x \) is 4 units, \( y \) is 5 units. So \( y = \frac{5}{4}x = 1.25x \), which is a 25% increase (since \( 1.25 - 1 = 0.25 \)). Wait, but 25% increase is \( y = 1.25x \), which is \( \frac{5}{4}x \). So Diagram B corresponds to a 25% increase? Wait, no, wait: 25% increase: original is \( x \), new is \( x + 0.25x = 1.25x \). Diagram B: \( x \) is 4, \( y \) is 5. So \( 5 = 1.25 \times 4 \), yes. So "An increase by 25%" matches Diagram B? Wait, no, wait the first question: "An increase by 25%". So the answer for "An increase by 25%" is Diagram B? Wait, no, let's check again.

Wait, Diagram A: \( y \) is shorter than \( x \), so decrease. Diagram B: \( y \) is longer than \( x \), so increase. So "An increase by 25%" would be Diagram B? Wait, but let's check the second part: "A decrease by 25%". Diagram A: \( y = 0.75x \), which is a 25% decrease (since \( 1 - 0.75 = 0.25 \)). So "A decrease by 25%" matches Diagram A? Wait, no, Diagram A: \( x \) is 4, \( y \) is 3. So \( 3 = 0.75 \times 4 \), so that's a 25% decrease. So "A decrease by 25%" matches Diagram A? Wait, the options are Diagram A, Diagram B, or neither.

Wait, let's re-express:

• Diagram A: \( x \) (original) has 4 equal parts, \( y \) (new) has 3 parts. So \( y = \frac{3}{4}x = 0.75x \). So this is a 25% decrease (since \( 1 - 0.75 = 0.25 \), so 25% decrease).

• Diagram B: \( x \) (original) has 4 equal parts, \( y \) (new) has 5 parts. So \( y = \frac{5}{4}x = 1.25x \). So this is a 25% increase (since \( 1.25 - 1 = 0.25 \), so 25% increase).

Now, the first question: "An increase by 25%". So \( y = 1.25x \), which is Diagram B. So answer: Diagram B.

Second question: "A decrease by 25%". \( y = 0.75x \), which is Diagram A. So answer: Diagram A.

Third question: \( y = 1.\overline{6}x \). \( 1.\overline{6} = \frac{5}{3} \approx 1.666... \). Diagram A: \( y = 0.75x \), Diagram B: \( y = 1.25x \). Neither is \( \frac{5}{3}x \), so answer: Neither.

2. A decrease by 25%

As above, Diagram A has \( y = 0.75x \), which is a 25% decrease (since \( 1 - 0.75 = 0.25 \)). So this matches Diagram A.

3. \( y = 1.\overline{6}x \)

\( 1.\overline{6} = \frac{5}{3} \approx 1.666... \). Diagram A: \( y = 0.75x \), Diagram B: \( y = 1.25x \). Neither is \( \frac{5}{3}x \), so it matches neither.

Answer:

s:

1. An increase by 25%: Diagram B
2. A decrease by 25%: Diagram A
3. \( y = 1.\overline{6}x \): Neither

(Note: For each sub-question, the answer is as follows. If the problem expects each part to be answered separately, here are the answers:

• An increase by 25%: Diagram B
• A decrease by 25%: Diagram A
• \( y = 1.\overline{6}x \): Neither)