Question

1. two middle school classes work together to paint a mural in their school. match each situation with the expression that could be used to answer the question. mr. perez’s class paints $\frac{1}{5}$ of the entire mural, which is $\frac{2}{3}$ of the section they must paint. what portion of the entire mural is their section? $\frac{2}{3}+\frac{1}{5}$ mrs. chan’s class must paint $\frac{1}{5}$ of the mural. the class paints $\frac{2}{3}$ of their section. what portion of the entire mural does mrs. chan’s class paint? $\frac{2}{3}-\frac{1}{5}$ mrs. chan’s class uses $\frac{2}{3}$ of a gallon of white paint. mr. perez’s class uses $\frac{1}{5}$ of the gallon of white paint. how much more paint does mrs. chan’s class use than mr. perez’s class? $\frac{1}{5}×\frac{2}{3}$ mrs. chan’s class uses $\frac{2}{3}$ of a gallon of white paint. mr. perez’s class uses $\frac{1}{5}$ of the gallon of white paint. what portion of the gallon of white paint do the classes use altogether? $\frac{1}{5}÷\frac{2}{3}$ 2. leo’s backpack weighs $12\frac{3}{4}$ pounds. sana’s backpack weighs $8\frac{1}{2}$ pounds. complete each statement. a. together, leo’s and sana’s backpacks weigh ____ pounds. b. leo’s backpack weighs __ pounds more than sana’s backpack. c. leo’s backpack weighs ____ times as much as sana’s backpack.

Explanation:

Problem 1 (Matching)

Mr. Perez’s class situation:

We know that the portion they painted ($\frac{1}{5}$ of the entire mural) is $\frac{2}{3}$ of their section. To find their section (portion of the entire mural), we need to find the number that when multiplied by $\frac{2}{3}$ gives $\frac{1}{5}$. So we use division: $\frac{1}{5}\div\frac{2}{3}$.

Mrs. Chan’s class (painting portion):

Mrs. Chan’s class must paint $\frac{1}{5}$ of the mural, and they painted $\frac{2}{3}$ of their section. To find the portion of the entire mural they painted, we multiply the portion of the mural they must paint by the portion they completed: $\frac{1}{5}\times\frac{2}{3}$.

Paint usage difference:

Mrs. Chan’s class uses $\frac{2}{3}$ gallon, Mr. Perez’s uses $\frac{1}{5}$ gallon. To find how much more, we subtract: $\frac{2}{3}-\frac{1}{5}$.

Total paint usage:

To find the total paint used, we add the amounts used by both classes: $\frac{2}{3}+\frac{1}{5}$.

Problem 2

Part a:

To find the total weight, we add Leo’s and Sana’s backpack weights.
Leo’s weight: $12\frac{3}{4}=\frac{12\times4 + 3}{4}=\frac{51}{4}$
Sana’s weight: $8\frac{1}{2}=\frac{8\times2+ 1}{2}=\frac{17}{2}=\frac{34}{4}$
Total: $\frac{51}{4}+\frac{34}{4}=\frac{51 + 34}{4}=\frac{85}{4}=21\frac{1}{4}$

Part b:

To find how much more Leo’s backpack weighs, we subtract Sana’s weight from Leo’s.
$\frac{51}{4}-\frac{34}{4}=\frac{51-34}{4}=\frac{17}{4}=4\frac{1}{4}$

Part c:

To find how many times Leo’s backpack weighs as much as Sana’s, we divide Leo’s weight by Sana’s.
Leo’s weight: $\frac{51}{4}$, Sana’s weight: $\frac{17}{2}$
$\frac{51}{4}\div\frac{17}{2}=\frac{51}{4}\times\frac{2}{17}=\frac{51\times2}{4\times17}=\frac{102}{68}=\frac{3}{2}=1\frac{1}{2}$

Final Answers

Problem 1 Matches:

• Mr. Perez’s class: $\boldsymbol{\frac{1}{5}\div\frac{2}{3}}$
• Mrs. Chan’s painting portion: $\boldsymbol{\frac{1}{5}\times\frac{2}{3}}$
• Paint usage difference: $\boldsymbol{\frac{2}{3}-\frac{1}{5}}$
• Total paint usage: $\boldsymbol{\frac{2}{3}+\frac{1}{5}}$

Problem 2:

a. $\boldsymbol{21\frac{1}{4}}$
b. $\boldsymbol{4\frac{1}{4}}$
c. $\boldsymbol{1\frac{1}{2}}$

Answer:

Problem 1 (Matching)

Mr. Perez’s class situation:

We know that the portion they painted ($\frac{1}{5}$ of the entire mural) is $\frac{2}{3}$ of their section. To find their section (portion of the entire mural), we need to find the number that when multiplied by $\frac{2}{3}$ gives $\frac{1}{5}$. So we use division: $\frac{1}{5}\div\frac{2}{3}$.

Mrs. Chan’s class (painting portion):

Mrs. Chan’s class must paint $\frac{1}{5}$ of the mural, and they painted $\frac{2}{3}$ of their section. To find the portion of the entire mural they painted, we multiply the portion of the mural they must paint by the portion they completed: $\frac{1}{5}\times\frac{2}{3}$.

Paint usage difference:

Mrs. Chan’s class uses $\frac{2}{3}$ gallon, Mr. Perez’s uses $\frac{1}{5}$ gallon. To find how much more, we subtract: $\frac{2}{3}-\frac{1}{5}$.

Total paint usage:

To find the total paint used, we add the amounts used by both classes: $\frac{2}{3}+\frac{1}{5}$.

Problem 2

Part a:

To find the total weight, we add Leo’s and Sana’s backpack weights.
Leo’s weight: $12\frac{3}{4}=\frac{12\times4 + 3}{4}=\frac{51}{4}$
Sana’s weight: $8\frac{1}{2}=\frac{8\times2+ 1}{2}=\frac{17}{2}=\frac{34}{4}$
Total: $\frac{51}{4}+\frac{34}{4}=\frac{51 + 34}{4}=\frac{85}{4}=21\frac{1}{4}$

Part b:

To find how much more Leo’s backpack weighs, we subtract Sana’s weight from Leo’s.
$\frac{51}{4}-\frac{34}{4}=\frac{51-34}{4}=\frac{17}{4}=4\frac{1}{4}$

Part c:

To find how many times Leo’s backpack weighs as much as Sana’s, we divide Leo’s weight by Sana’s.
Leo’s weight: $\frac{51}{4}$, Sana’s weight: $\frac{17}{2}$
$\frac{51}{4}\div\frac{17}{2}=\frac{51}{4}\times\frac{2}{17}=\frac{51\times2}{4\times17}=\frac{102}{68}=\frac{3}{2}=1\frac{1}{2}$

Final Answers

Problem 1 Matches:

• Mr. Perez’s class: $\boldsymbol{\frac{1}{5}\div\frac{2}{3}}$
• Mrs. Chan’s painting portion: $\boldsymbol{\frac{1}{5}\times\frac{2}{3}}$
• Paint usage difference: $\boldsymbol{\frac{2}{3}-\frac{1}{5}}$
• Total paint usage: $\boldsymbol{\frac{2}{3}+\frac{1}{5}}$

Problem 2:

a. $\boldsymbol{21\frac{1}{4}}$
b. $\boldsymbol{4\frac{1}{4}}$
c. $\boldsymbol{1\frac{1}{2}}$