Question

1. carlos gonzales was a left fielder for the colorado rockies, a major league baseball team. he played 135 games in 2012 and hit 22 home runs. if carlos played a total of 1,075 games in his career, which proportion could you solve to determine the approximate number of homeruns he hit? assume he hit homeruns at the same rate throughout his career. in these proportions, the variable ( h ) represents the approximate number of homeruns. ( \frac{135}{22} = \frac{1075}{h} ) ( \frac{135}{1075} = \frac{h}{22} ) ( \frac{135}{22} = \frac{h}{1075} ) ( \frac{22}{135} = \frac{1075}{h} )

Explanation:

Step1: Define the ratio

The rate of home runs per game should be constant. In 2012, games = 135, home runs = 22. Career games = 1075, career home runs = \( h \). So the ratio of games to home runs in 2012 should equal the ratio of career games to career home runs.

Step2: Set up the proportion

In 2012: \( \frac{\text{games}}{\text{home runs}} = \frac{135}{22} \). Career: \( \frac{\text{games}}{\text{home runs}} = \frac{1075}{h} \). Since rates are equal, \( \frac{135}{22} = \frac{1075}{h} \) is incorrect. Wait, no—wait, the rate of home runs per game is \( \frac{\text{home runs}}{\text{games}} \). So 2012 rate: \( \frac{22}{135} \), career rate: \( \frac{h}{1075} \). Wait, no, the problem says "hit homeruns at the same rate", so \( \frac{\text{home runs}}{\text{games}} \) is constant. So \( \frac{22}{135} = \frac{h}{1075} \)? Wait, no, let's re-express. Wait, the options: let's check each.

Option 1: \( \frac{135}{22} = \frac{1075}{h} \) → cross-multiply: \( 135h = 22×1075 \) → \( h = \frac{22×1075}{135} \). But the rate here is games per home run. If the rate (games per home run) is constant, then 135 games / 22 home runs = 1075 games / h home runs. That makes sense. Wait, but let's check the correct proportion. The rate of home runs per game: 22 home runs in 135 games, so per game: \( \frac{22}{135} \). In career, h home runs in 1075 games, so per game: \( \frac{h}{1075} \). Setting equal: \( \frac{22}{135} = \frac{h}{1075} \), which is equivalent to \( \frac{135}{22} = \frac{1075}{h} \) (cross-multiplied: \( 22×1075 = 135h \) vs \( 135h = 22×1075 \), same thing). Wait, no, cross-multiplying \( \frac{22}{135} = \frac{h}{1075} \) gives \( 22×1075 = 135h \), and \( \frac{135}{22} = \frac{1075}{h} \) also gives \( 135h = 22×1075 \), same equation. Wait, but let's check the fourth option: \( \frac{22}{135} = \frac{1075}{h} \). Wait, no, the fourth option is \( \frac{22}{135} = \frac{1075}{h} \)? Wait, the fourth option is \( \frac{22}{135} = \frac{1075}{h} \)? Wait, the options are:

1. \( \frac{135}{22} = \frac{1075}{h} \)

2. \( \frac{135}{1075} = \frac{h}{22} \)

3. \( \frac{135}{22} = \frac{h}{1075} \)

4. \( \frac{22}{135} = \frac{1075}{h} \)

Wait, let's re-express the correct proportion. The key is that the ratio of home runs to games is constant. So \( \frac{\text{home runs}}{\text{games}} = \frac{22}{135} = \frac{h}{1075} \). Cross-multiplying, \( 22×1075 = 135h \), which is the same as \( \frac{22}{135} = \frac{h}{1075} \), but that's not an option. Wait, the options: let's check option 4: \( \frac{22}{135} = \frac{1075}{h} \) → cross-multiply: \( 22h = 135×1075 \) → \( h = \frac{135×1075}{22} \), which is wrong. Wait, I must have messed up. Let's start over.

Carlos hit 22 home runs in 135 games. So the ratio of home runs to games is \( \frac{22}{135} \). In his career, he hit \( h \) home runs in 1075 games, so the ratio is \( \frac{h}{1075} \). Since the rate (home runs per game) is constant, \( \frac{22}{135} = \frac{h}{1075} \). Now, let's rearrange this equation. Cross-multiplying: \( 22×1075 = 135h \). Now, let's look at the options:

Option 1: \( \frac{135}{22} = \frac{1075}{h} \) → cross-multiplying: \( 135h = 22×1075 \), which is the same as \( 22×1075 = 135h \). So this is equivalent. So option 1 is correct? Wait, no, \( \frac{135}{22} \) is games per home run, and \( \frac{1075}{h} \) is also games per home run. So if the number of games per home run is constant, then 135 games / 22 home runs = 1075 games / h home runs. That makes sense. So option 1 is \( \frac{135}{22} = \frac{1075}{h} \), which is correct. Wait, but let's check the fourth option: \( \frac{22}{135} = \frac{1075}{h} \). That would be home runs per game = games per home run, which is not correct. So the correct proportion is option 1? Wait, no, wait the fourth option is \( \frac{22}{135} = \frac{1075}{h} \)? Wait, the fourth option is \( \frac{22}{135} = \frac{1075}{h} \)? Let me check the original options again.

The options are:

1. \( \frac{135}{22} = \frac{1075}{h} \)

2. \( \frac{135}{1075} = \frac{h}{22} \)

3. \( \frac{135}{22} = \frac{h}{1075} \)

4. \( \frac{22}{135} = \frac{1075}{h} \)

Wait, option 4 is \( \frac{22}{135} = \frac{1075}{h} \). Let's solve for h in each:

Option 1: \( 135h = 22×1075 \) → \( h = (22×1075)/135 \)

Option 2: \( 135×22 = 1075h \) → \( h = (135×22)/1075 \) (too small, since career games are more than 135, h should be more than 22)

Option 3: \( 135×1075 = 22h \) → \( h = (135×1075)/22 \) (too big, since 135×1075 is way larger than 22×22)

Option 4: \( 22h = 135×1075 \) → \( h = (135×1075)/22 \) (same as option 3, which is wrong)

Wait, I must have made a mistake. Wait, the problem says "hit homeruns at the same rate throughout his career". The rate is home runs per game, so 22 home runs in 135 games, so per game: 22/135. In career, h home runs in 1075 games, so per game: h/1075. So 22/135 = h/1075. Now, let's rearrange this: cross-multiplying, 22×1075 = 135h → h = (22×1075)/135. Now, let's check the options:

Option 1: 135/22 = 1075/h → cross-multiplying: 135h = 22×1075 → h = (22×1075)/135. Yes! So option 1 is correct. Because 135/22 is games per home run (135 games for 22 home runs), and 1075/h is games per home run (1075 games for h home runs). Since the rate (games per home run) is constant, these are equal. So option 1 is correct. Wait, but earlier I thought option 4, but no—option 1 is \( \frac{135}{22} = \frac{1075}{h} \), which is correct.

Wait, let's verify with numbers. 22 home runs in 135 games: games per home run ≈ 6.136. Career games: 1075. So h = 1075 / 6.136 ≈ 175. Let's compute (22×1075)/135: 22×1075 = 23650; 23650 / 135 ≈ 175.18, which is reasonable (more than 22, since more games). Now, option 4: (135×1075)/22 ≈ (145125)/22 ≈ 6596, which is impossible. So option 1 is correct.

Answer:

A. \( \frac{135}{22} = \frac{1075}{h} \)