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Step-by-step solution for: Untitled
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Show Answer Key & Explanations
Step-by-step solution for: Untitled
Let’s solve each problem one by one. We’ll use simple algebra — just setting up equations based on the words, then solving them step by step.
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Problem 1:
*The second of two numbers is 4 times the first. Their sum is 50. Find the numbers.*
Let the first number = x
Then the second number = 4x
Sum: x + 4x = 50 → 5x = 50 → x = 10
So numbers are 10 and 40
Look at answer choices: E says “10, 40” → So letter for problem 1 is E
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Problem 2:
*The larger of two numbers is 12 more than the smaller. Their sum is 84. Find the numbers.*
Let smaller = x
Larger = x + 12
Sum: x + (x + 12) = 84 → 2x + 12 = 84 → 2x = 72 → x = 36
Numbers: 36 and 48
Answer choice T: “36, 48” → Letter for problem 2 is T
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Problem 3:
*The sum of two numbers is 45. The first is 9 less than the second. Find the numbers.*
Let second = x
First = x - 9
Sum: (x - 9) + x = 45 → 2x - 9 = 45 → 2x = 54 → x = 27
Numbers: 18 and 27
Answer choice S: “18, 27” → Letter for problem 3 is S
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Problem 4:
*The second of two numbers is 5 more than twice the first. Their sum is 80. Find the numbers.*
Let first = x
Second = 2x + 5
Sum: x + (2x + 5) = 80 → 3x + 5 = 80 → 3x = 75 → x = 25
Numbers: 25 and 55
Answer choice O: “25, 55” → Letter for problem 4 is O
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Problem 5:
*The larger of two numbers is 1 less than 3 times the smaller. Their sum is 63. Find the numbers.*
Let smaller = x
Larger = 3x - 1
Sum: x + (3x - 1) = 63 → 4x - 1 = 63 → 4x = 64 → x = 16
Numbers: 16 and 47
Answer choice I: “16, 47” → Letter for problem 5 is I
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Problem 6:
*Find two numbers whose sum is 92, if the first is 4 more than 7 times the second.*
Let second = x
First = 7x + 4
Sum: (7x + 4) + x = 92 → 8x + 4 = 92 → 8x = 88 → x = 11
Numbers: 7(11)+4 = 81, and 11 → 81, 11
Answer choice G: “81, 11” → Letter for problem 6 is G
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Problem 7:
*The sum of two numbers is 172. The first is 8 less than 5 times the second. Find the first number.*
Let second = x
First = 5x - 8
Sum: (5x - 8) + x = 172 → 6x - 8 = 172 → 6x = 180 → x = 30
First number = 5(30) - 8 = 150 - 8 = 142
Answer choice H: “142” → Letter for problem 7 is H
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Problem 8:
*Together, a necklace and a bracelet cost $192. Necklace costs 3 times as much as bracelet. Find price of each.*
Let bracelet = x
Necklace = 3x
Total: x + 3x = 192 → 4x = 192 → x = 48
Bracelet = $48, Necklace = $144
Answer choice R: “$48, $144” → Letter for problem 8 is R
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Problem 9:
*Grandpa’s age is 6 years less than 6 times Junior’s age. Sum of their ages is 78. Find each age.*
Let Junior’s age = x
Grandpa = 6x - 6
Sum: x + (6x - 6) = 78 → 7x - 6 = 78 → 7x = 84 → x = 12
Junior = 12, Grandpa = 6(12)-6 = 72-6 = 66
Answer choice N: “12, 66” → Letter for problem 9 is N
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Problem 10:
*First film lasted 3 minutes less than twice as long as the second. Together they lasted 132 minutes. How long was the first film?*
Let second film = x minutes
First film = 2x - 3
Total: (2x - 3) + x = 132 → 3x - 3 = 132 → 3x = 135 → x = 45
First film = 2(45) - 3 = 90 - 3 = 87 minutes
Answer choice L: “87 min” → Letter for problem 10 is L
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Now we have all letters:
Problem 1 → E
Problem 2 → T
Problem 3 → S
Problem 4 → O
Problem 5 → I
Problem 6 → G
Problem 7 → H
Problem 8 → R
Problem 9 → N
Problem 10 → L
Now look at the bottom grid:
It has boxes with numbers:
8 | 4 | 10 | 10 | 5 | 9 | 6 | 5 | 9 | 2 | 7 | 1 | 5 | 3 | 10 | 1 | 3
We need to write the letter corresponding to each problem number in the box that contains that problem number.
Wait — let me re-read the instructions:
> "Write this letter in each box that contains the number of that problem."
That means: For each problem (say problem 1), find every box in the bottom row that has the number “1”, and put the letter for problem 1 in those boxes.
But looking at the bottom row, it's a sequence of numbers from 1 to 10, repeated? Let’s list them with positions:
Position: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Number: 8 4 10 10 5 9 6 5 9 2 7 1 5 3 10 1 3
So:
- Boxes with “1”: positions 12 and 16 → put letter for problem 1 → E
- Boxes with “2”: position 10 → put letter for problem 2 → T
- Boxes with “3”: positions 14 and 17 → put letter for problem 3 → S
- Boxes with “4”: position 2 → put letter for problem 4 → O
- Boxes with “5”: positions 5, 8, 13 → put letter for problem 5 → I
- Boxes with “6”: position 7 → put letter for problem 6 → G
- Boxes with “7”: position 11 → put letter for problem 7 → H
- Boxes with “8”: position 1 → put letter for problem 8 → R
- Boxes with “9”: positions 6 and 9 → put letter for problem 9 → N
- Boxes with “10”: positions 3, 4, 15 → put letter for problem 10 → L
Now fill the grid:
Pos 1 (num 8) → R
Pos 2 (num 4) → O
Pos 3 (num 10) → L
Pos 4 (num 10) → L
Pos 5 (num 5) → I
Pos 6 (num 9) → N
Pos 7 (num 6) → G
Pos 8 (num 5) → I
Pos 9 (num 9) → N
Pos 10 (num 2) → T
Pos 11 (num 7) → H
Pos 12 (num 1) → E
Pos 13 (num 5) → I
Pos 14 (num 3) → S
Pos 15 (num 10) → L
Pos 16 (num 1) → E
Pos 17 (num 3) → S
So the filled grid reads:
R O L L I N G I N T H E I S L E S
Wait — let’s write it out:
Positions 1 to 17:
1: R
2: O
3: L
4: L
5: I
6: N
7: G
8: I
9: N
10: T
11: H
12: E
13: I
14: S
15: L
16: E
17: S
So: R O L L I N G I N T H E I S L E S
Grouped: ROLLING IN THE ISLES
Ah! That’s the punchline: “What do they call bowling in Hawaii?” → “Rolling in the Isles” (a play on “rolling in the aisles” meaning laughing hard, but here it’s “isles” like Hawaiian islands).
Perfect!
Final Answer:
ROLLING IN THE ISLES
---
Problem 1:
*The second of two numbers is 4 times the first. Their sum is 50. Find the numbers.*
Let the first number = x
Then the second number = 4x
Sum: x + 4x = 50 → 5x = 50 → x = 10
So numbers are 10 and 40
Look at answer choices: E says “10, 40” → So letter for problem 1 is E
---
Problem 2:
*The larger of two numbers is 12 more than the smaller. Their sum is 84. Find the numbers.*
Let smaller = x
Larger = x + 12
Sum: x + (x + 12) = 84 → 2x + 12 = 84 → 2x = 72 → x = 36
Numbers: 36 and 48
Answer choice T: “36, 48” → Letter for problem 2 is T
---
Problem 3:
*The sum of two numbers is 45. The first is 9 less than the second. Find the numbers.*
Let second = x
First = x - 9
Sum: (x - 9) + x = 45 → 2x - 9 = 45 → 2x = 54 → x = 27
Numbers: 18 and 27
Answer choice S: “18, 27” → Letter for problem 3 is S
---
Problem 4:
*The second of two numbers is 5 more than twice the first. Their sum is 80. Find the numbers.*
Let first = x
Second = 2x + 5
Sum: x + (2x + 5) = 80 → 3x + 5 = 80 → 3x = 75 → x = 25
Numbers: 25 and 55
Answer choice O: “25, 55” → Letter for problem 4 is O
---
Problem 5:
*The larger of two numbers is 1 less than 3 times the smaller. Their sum is 63. Find the numbers.*
Let smaller = x
Larger = 3x - 1
Sum: x + (3x - 1) = 63 → 4x - 1 = 63 → 4x = 64 → x = 16
Numbers: 16 and 47
Answer choice I: “16, 47” → Letter for problem 5 is I
---
Problem 6:
*Find two numbers whose sum is 92, if the first is 4 more than 7 times the second.*
Let second = x
First = 7x + 4
Sum: (7x + 4) + x = 92 → 8x + 4 = 92 → 8x = 88 → x = 11
Numbers: 7(11)+4 = 81, and 11 → 81, 11
Answer choice G: “81, 11” → Letter for problem 6 is G
---
Problem 7:
*The sum of two numbers is 172. The first is 8 less than 5 times the second. Find the first number.*
Let second = x
First = 5x - 8
Sum: (5x - 8) + x = 172 → 6x - 8 = 172 → 6x = 180 → x = 30
First number = 5(30) - 8 = 150 - 8 = 142
Answer choice H: “142” → Letter for problem 7 is H
---
Problem 8:
*Together, a necklace and a bracelet cost $192. Necklace costs 3 times as much as bracelet. Find price of each.*
Let bracelet = x
Necklace = 3x
Total: x + 3x = 192 → 4x = 192 → x = 48
Bracelet = $48, Necklace = $144
Answer choice R: “$48, $144” → Letter for problem 8 is R
---
Problem 9:
*Grandpa’s age is 6 years less than 6 times Junior’s age. Sum of their ages is 78. Find each age.*
Let Junior’s age = x
Grandpa = 6x - 6
Sum: x + (6x - 6) = 78 → 7x - 6 = 78 → 7x = 84 → x = 12
Junior = 12, Grandpa = 6(12)-6 = 72-6 = 66
Answer choice N: “12, 66” → Letter for problem 9 is N
---
Problem 10:
*First film lasted 3 minutes less than twice as long as the second. Together they lasted 132 minutes. How long was the first film?*
Let second film = x minutes
First film = 2x - 3
Total: (2x - 3) + x = 132 → 3x - 3 = 132 → 3x = 135 → x = 45
First film = 2(45) - 3 = 90 - 3 = 87 minutes
Answer choice L: “87 min” → Letter for problem 10 is L
---
Now we have all letters:
Problem 1 → E
Problem 2 → T
Problem 3 → S
Problem 4 → O
Problem 5 → I
Problem 6 → G
Problem 7 → H
Problem 8 → R
Problem 9 → N
Problem 10 → L
Now look at the bottom grid:
It has boxes with numbers:
8 | 4 | 10 | 10 | 5 | 9 | 6 | 5 | 9 | 2 | 7 | 1 | 5 | 3 | 10 | 1 | 3
We need to write the letter corresponding to each problem number in the box that contains that problem number.
Wait — let me re-read the instructions:
> "Write this letter in each box that contains the number of that problem."
That means: For each problem (say problem 1), find every box in the bottom row that has the number “1”, and put the letter for problem 1 in those boxes.
But looking at the bottom row, it's a sequence of numbers from 1 to 10, repeated? Let’s list them with positions:
Position: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17
Number: 8 4 10 10 5 9 6 5 9 2 7 1 5 3 10 1 3
So:
- Boxes with “1”: positions 12 and 16 → put letter for problem 1 → E
- Boxes with “2”: position 10 → put letter for problem 2 → T
- Boxes with “3”: positions 14 and 17 → put letter for problem 3 → S
- Boxes with “4”: position 2 → put letter for problem 4 → O
- Boxes with “5”: positions 5, 8, 13 → put letter for problem 5 → I
- Boxes with “6”: position 7 → put letter for problem 6 → G
- Boxes with “7”: position 11 → put letter for problem 7 → H
- Boxes with “8”: position 1 → put letter for problem 8 → R
- Boxes with “9”: positions 6 and 9 → put letter for problem 9 → N
- Boxes with “10”: positions 3, 4, 15 → put letter for problem 10 → L
Now fill the grid:
Pos 1 (num 8) → R
Pos 2 (num 4) → O
Pos 3 (num 10) → L
Pos 4 (num 10) → L
Pos 5 (num 5) → I
Pos 6 (num 9) → N
Pos 7 (num 6) → G
Pos 8 (num 5) → I
Pos 9 (num 9) → N
Pos 10 (num 2) → T
Pos 11 (num 7) → H
Pos 12 (num 1) → E
Pos 13 (num 5) → I
Pos 14 (num 3) → S
Pos 15 (num 10) → L
Pos 16 (num 1) → E
Pos 17 (num 3) → S
So the filled grid reads:
R O L L I N G I N T H E I S L E S
Wait — let’s write it out:
Positions 1 to 17:
1: R
2: O
3: L
4: L
5: I
6: N
7: G
8: I
9: N
10: T
11: H
12: E
13: I
14: S
15: L
16: E
17: S
So: R O L L I N G I N T H E I S L E S
Grouped: ROLLING IN THE ISLES
Ah! That’s the punchline: “What do they call bowling in Hawaii?” → “Rolling in the Isles” (a play on “rolling in the aisles” meaning laughing hard, but here it’s “isles” like Hawaiian islands).
Perfect!
Final Answer:
ROLLING IN THE ISLES
Parent Tip: Review the logic above to help your child master the concept of algebra with pizzazz page 40.