Question

lesson 1 practice problems

1. tyler reads \\(\frac{2}{15}\\) of a book on monday, \\(\frac{1}{3}\\) of it on tuesday, \\(\frac{2}{9}\\) of it on wednesday, and \\(\frac{3}{4}\\) of the remainder on thursday. if he still has 14 pages left to read on friday, how many pages are there in the book?

1. clare asks andre to play the following number puzzle:

• pick a number
• add 2
• multiply by 3
• subtract 7
• add your original number

andre’s final result is 27.
which number did he start with?

Explanation:

Problem 1

Step 1: Find the fraction read by Wednesday

First, we need to find the total fraction of the book read by Wednesday. The fractions read on Monday, Tuesday, and Wednesday are $\frac{2}{15}$, $\frac{1}{3}$, and $\frac{2}{9}$ respectively. We need to find a common denominator, which is 45.

• $\frac{2}{15}=\frac{2\times3}{15\times3}=\frac{6}{45}$
• $\frac{1}{3}=\frac{1\times15}{3\times15}=\frac{15}{45}$
• $\frac{2}{9}=\frac{2\times5}{9\times5}=\frac{10}{45}$

Now, add these fractions: $\frac{6}{45}+\frac{15}{45}+\frac{10}{45}=\frac{6 + 15+10}{45}=\frac{31}{45}$

Step 2: Find the remainder after Wednesday

The remainder after Wednesday is $1-\frac{31}{45}=\frac{45 - 31}{45}=\frac{14}{45}$

Step 3: Find the fraction read on Thursday

On Thursday, he reads $\frac{3}{4}$ of the remainder. So the fraction read on Thursday is $\frac{3}{4}\times\frac{14}{45}=\frac{42}{180}=\frac{7}{30}$ (simplified by dividing numerator and denominator by 6)

Step 4: Find the remaining fraction on Friday

The remaining fraction on Friday is the remainder after Wednesday minus the fraction read on Thursday. So:
$\frac{14}{45}-\frac{7}{30}$
Find a common denominator, which is 90.

• $\frac{14}{45}=\frac{14\times2}{45\times2}=\frac{28}{90}$
• $\frac{7}{30}=\frac{7\times3}{30\times3}=\frac{21}{90}$

Subtract: $\frac{28}{90}-\frac{21}{90}=\frac{7}{90}$

Step 5: Find the total number of pages

Let the total number of pages be $x$. We know that $\frac{7}{90}x = 14$ (since the remaining pages are 14, which is $\frac{7}{90}$ of the total book).
To solve for $x$, we can multiply both sides by $\frac{90}{7}$:
$x=14\times\frac{90}{7}=2\times90 = 180$

Problem 2

Step 1: Define the variable

Let the number Andre started with be $x$.

Step 2: Translate the operations into an equation

• Pick a number: $x$
• Add 2: $x + 2$
• Multiply by 3: $3(x + 2)$
• Subtract 7: $3(x + 2)-7$
• Add the original number: $3(x + 2)-7+x$

We know the final result is 27, so the equation is:
$3(x + 2)-7+x=27$

Step 3: Simplify and solve the equation

First, expand the equation:
$3x+6 - 7+x=27$
Combine like terms:
$(3x+x)+(6 - 7)=27$
$4x-1 = 27$

Add 1 to both sides:
$4x=27 + 1=28$

Divide both sides by 4:
$x=\frac{28}{4}=7$

Answer:

s:

1. The book has $\boldsymbol{180}$ pages.
2. Andre started with the number $\boldsymbol{7}$.