Question

the number line models the expression 476 - 289. enter numbers to label each jump; first to the next 100, then to the difference of 476 and 289.

Explanation:

Step1: Find the first jump (to next 100)

We start from 476 and jump back to 400. The distance of this jump is \( 476 - 400 = 76 \). Wait, no, wait. Wait, the problem says "first to the next 100" – wait, actually, we are doing \( 476 - 289 \). Let's see the number line: the rightmost point is 476, then we jump left to 400 (that's a jump of \( 476 - 400 = 76 \)? Wait, no, maybe I got it reversed. Wait, the number line has 400 and 476, with an arrow from 476 to 400 (the first red arc), then from 400 to the left (the second red arc), and then the leftmost box. Wait, the expression is \( 476 - 289 \). Let's compute \( 476 - 289 \). Let's break down 289: 289 is 200 + 80 + 9? No, the problem says "first to the next 100" – wait, maybe the first jump is from 476 to 400? Wait, 476 to 400: the difference is \( 476 - 400 = 76 \)? No, wait, 400 is less than 476, so jumping from 476 to 400 is a left jump of 76? Wait, no, maybe the first jump is from 400 to the left to reach \( 400 - x \), and then the total is \( 476 - 289 \). Wait, let's compute \( 476 - 289 \). Let's do 476 - 200 = 276, 276 - 80 = 196, 196 - 9 = 187. But the number line approach: first, from 476, jump back to 400 (that's a jump of \( 476 - 400 = 76 \))? Wait, no, the problem says "first to the next 100" – wait, maybe the first jump is from 476 to 400 (the next lower 100), so that jump is \( 476 - 400 = 76 \)? Wait, no, the problem says "label each jump; first to the next 100, then to the difference of 476 and 289". Wait, maybe the first jump (from 476 to 400) is 76? Wait, no, let's re-express \( 289 \) as \( 200 + 89 \)? No, the number line has 400, so let's split \( 289 \) into \( 200 + 89 \)? Wait, no, \( 476 - 289 = 476 - (200 + 89) = (476 - 200) - 89 = 276 - 89 \). But the number line shows 476 -> 400 (jump of 76), then 400 ->? Let's compute \( 476 - 289 = 187 \). So \( 400 - 187 = 213 \)? No, wait, maybe the first jump (from 476 to 400) is \( 476 - 400 = 76 \), then the second jump (from 400 to the left) is \( 400 - (476 - 289) = 400 - 187 = 213 \)? No, that doesn't make sense. Wait, let's do it step by step.

First, the rightmost point is 476. The first jump (red arc from 476 to 400) is \( 476 - 400 = 76 \). Then, we need to jump from 400 to the left to reach \( 476 - 289 = 187 \). So the distance from 400 to 187 is \( 400 - 187 = 213 \)? Wait, no, \( 400 - 213 = 187 \), yes. So the first jump (from 476 to 400) is 76, the second jump (from 400 to 187) is 213, and the leftmost box is 187 (since \( 476 - 289 = 187 \)). Wait, let's check: \( 76 + 213 = 289 \), and \( 476 - 289 = 187 \). Yes, that works. So:

- The jump from 476 to 400: \( 476 - 400 = 76 \) (so the box above that arc is 76? Wait, no, the problem says "label each jump; first to the next 100, then to the difference of 476 and 289". Wait, maybe the first jump is from 400 to the left (to reach the next lower 100? Wait, 400 - 200 = 200? No, 289 is 200 + 89. Wait, maybe I messed up the direction. Let's re-express \( 476 - 289 \) as \( 476 - 200 - 80 - 9 \), but the number line has 400. So 476 - 200 = 276, but 276 is more than 400? No, 276 is less than 400. Wait, no, 476 - 289: let's do 476 - 300 = 176, then add back 11 (since we subtracted 300 instead of 289, so 176 + 11 = 187). Ah, that's a better way: 289 is 300 - 11, so \( 476 - 289 = 476 - (300 - 11) = 476 - 300 + 11 = 176 + 11 = 187 \). Now, using the number line: from 476, jump back 300 to 176? No, the number line has 400. Wait, maybe the first jump is from 476 to 400 (jump of 76), then from 400 to 187 (jump of 213), because 76 + 213 = 289, and 476 - 289 = 187. So:

- The jump from 476 to 400: 76 (so the box above that arc is 76)
- The jump from 400 to 187: 213 (so the box above that arc is 213)
- The leftmost box: 187 (the result of \( 476 - 289 \))

Let's verify: \( 476 - 76 = 400 \), \( 400 - 213 = 187 \), and \( 76 + 213 = 289 \), so \( 476 - 289 = 187 \). Perfect.

Step2: Confirm the values

- Jump from 476 to 400: \( 476 - 400 = 76 \) (so the box above the first arc (476 to 400) is 76)
- Jump from 400 to 187: \( 400 - 187 = 213 \) (so the box above the second arc (400 to 187) is 213)
- Leftmost box: \( 476 - 289 = 187 \)

Wait, but the problem says "label each jump; first to the next 100, then to the difference of 476 and 289". Wait, maybe the "first jump" is from 400 to the left (to the next 100 below 400, which is 300? No, 400 - 100 = 300, but 289 is 200 + 89. Wait, maybe my initial approach was wrong. Let's re-express \( 289 \) as \( 200 + 89 \). Then \( 476 - 200 = 276 \), \( 276 - 89 = 187 \). But the number line has 400, so maybe the first jump is from 476 to 400 (76), then from 400 to 187 (213), because 76 + 213 = 289, and 476 - 289 = 187. So the labels are:

- The jump from 476 to 400: 76 (the box above that arc)
- The jump from 400 to 187: 213 (the box above that arc)
- The leftmost box: 187 (the result)

Wait, but let's check the arithmetic: \( 476 - 76 = 400 \), \( 400 - 213 = 187 \), and \( 76 + 213 = 289 \), so \( 476 - 289 = 187 \). Correct.

So:

- The first jump (from 476 to 400) is 76.
- The second jump (from 400 to the left) is 213.
- The leftmost box is 187.

Wait, but the problem says "label each jump; first to the next 100, then to the difference of 476 and 289". Wait, maybe "first to the next 100" means from 476, jump to the next lower 100, which is 400 (since 476 is between 400 and 500, the next lower 100 is 400). So that jump is 76 (476 - 400 = 76). Then, the remaining jump is 289 - 76 = 213 (since we need to subtract 289 total, and we already subtracted 76, so we need to subtract 213 more). Then the result is 476 - 289 = 187.

Yes, that makes sense.

Answer:

• The jump from 476 to 400: \( \boldsymbol{76} \)
• The jump from 400 to the left: \( \boldsymbol{213} \)
• The leftmost box (result): \( \boldsymbol{187} \)

(Assuming the boxes are: the first box above the 476->400 arc is 76, the second box above the 400->left arc is 213, and the leftmost box is 187.)