Question

4-2 copy and complete each of the diamond problems below. the pattern used in the diamond problems is shown at right. homework help
4-43. lucy keeps track of how long it takes her to do the newspaper’s crossword puzzle each day. her new times, in minutes, were: homework help

Explanation:

To solve Diamond Problems, we use the pattern: the top number is the sum of the right and left numbers, and the bottom number is the product of the right and left numbers. Let's solve each sub - problem:

Part (a)

• Let the left number be \(x\), the right number be \(6\), the top number (sum) is unknown, and the bottom number (product) is \(- 1\).
• We know that for the product: \(x\times6=-1\), so \(x =-\frac{1}{6}\).
• For the sum: The top number is \(x + 6=-\frac{1}{6}+6=\frac{- 1 + 36}{6}=\frac{35}{6}\).

Part (b)

• Let the right number be \(y\), the left number is \(-5\), the top number (sum) is \(3\), and the bottom number (product) is unknown.
• First, find \(y\) using the sum: \(-5 + y=3\), so \(y=3 + 5 = 8\).
• Then, find the product: The bottom number is \(-5\times8=-40\).

Part (c)

• Let the right number be \(z\), the left number is \(112\), the top number (sum) is \(12\), and the bottom number (product) is unknown.
• First, find \(z\) using the sum: \(112+z = 12\), so \(z=12 - 112=-100\).
• Then, find the product: The bottom number is \(112\times(-100)=-11200\).

Part (d)

• Let the left number be \(m\), the right number is \(24\), the top number (sum) is unknown, and the bottom number (product) is \(10\).
• First, find \(m\) using the product: \(m\times24 = 10\), so \(m=\frac{10}{24}=\frac{5}{12}\).
• Then, find the sum: The top number is \(m + 24=\frac{5}{12}+24=\frac{5+288}{12}=\frac{293}{12}\).

Final Answers

• (a) Top: \(\boldsymbol{\frac{35}{6}}\), Left: \(\boldsymbol{-\frac{1}{6}}\)
• (b) Right: \(\boldsymbol{8}\), Bottom: \(\boldsymbol{-40}\)
• (c) Right: \(\boldsymbol{-100}\), Bottom: \(\boldsymbol{-11200}\)
• (d) Left: \(\boldsymbol{\frac{5}{12}}\), Top: \(\boldsymbol{\frac{293}{12}}\)

Answer:

To solve Diamond Problems, we use the pattern: the top number is the sum of the right and left numbers, and the bottom number is the product of the right and left numbers. Let's solve each sub - problem:

Part (a)

• Let the left number be \(x\), the right number be \(6\), the top number (sum) is unknown, and the bottom number (product) is \(- 1\).
• We know that for the product: \(x\times6=-1\), so \(x =-\frac{1}{6}\).
• For the sum: The top number is \(x + 6=-\frac{1}{6}+6=\frac{- 1 + 36}{6}=\frac{35}{6}\).

Part (b)

• Let the right number be \(y\), the left number is \(-5\), the top number (sum) is \(3\), and the bottom number (product) is unknown.
• First, find \(y\) using the sum: \(-5 + y=3\), so \(y=3 + 5 = 8\).
• Then, find the product: The bottom number is \(-5\times8=-40\).

Part (c)

• Let the right number be \(z\), the left number is \(112\), the top number (sum) is \(12\), and the bottom number (product) is unknown.
• First, find \(z\) using the sum: \(112+z = 12\), so \(z=12 - 112=-100\).
• Then, find the product: The bottom number is \(112\times(-100)=-11200\).

Part (d)

• Let the left number be \(m\), the right number is \(24\), the top number (sum) is unknown, and the bottom number (product) is \(10\).
• First, find \(m\) using the product: \(m\times24 = 10\), so \(m=\frac{10}{24}=\frac{5}{12}\).
• Then, find the sum: The top number is \(m + 24=\frac{5}{12}+24=\frac{5+288}{12}=\frac{293}{12}\).

Final Answers

• (a) Top: \(\boldsymbol{\frac{35}{6}}\), Left: \(\boldsymbol{-\frac{1}{6}}\)
• (b) Right: \(\boldsymbol{8}\), Bottom: \(\boldsymbol{-40}\)
• (c) Right: \(\boldsymbol{-100}\), Bottom: \(\boldsymbol{-11200}\)
• (d) Left: \(\boldsymbol{\frac{5}{12}}\), Top: \(\boldsymbol{\frac{293}{12}}\)