Question

1. $3.2 = \frac{4}{5}(b - 5)$

$3.2 = (\frac{4}{5}\cdot\square) + (\frac{4}{5}\cdot\square)$
$3.2 = \square - 4$
$3.2 + \square = \square$
$\frac{5}{4}\cdot\square = \square$
$b = 9$

Explanation:

Step1: Distribute the fraction

We start with the equation \( 3.2=\frac{4}{5}(b - 5) \). First, we can rewrite \( 3.2 \) as a fraction. Since \( 3.2=\frac{32}{10}=\frac{16}{5} \), and also note that \( \frac{4}{5}\times5 = 4 \). But maybe it's easier to use the distributive property. The distributive property states that \( a(b - c)=ab - ac \). So applying that to \( \frac{4}{5}(b - 5) \), we get \( \frac{4}{5}b-\frac{4}{5}\times5 \). Calculating \( \frac{4}{5}\times5 \), the 5s cancel out, and we get 4. So the equation becomes \( 3.2=\frac{4}{5}b - 4 \).

Step2: Add 4 to both sides

To isolate the term with \( b \), we add 4 to both sides of the equation. So \( 3.2 + 4=\frac{4}{5}b-4 + 4 \). Calculating the left side: \( 3.2+4 = 7.2 \). The right side simplifies to \( \frac{4}{5}b \) because \( - 4+4 = 0 \). So now we have \( 7.2=\frac{4}{5}b \).

Step3: Solve for \( b \)

We know that \( \frac{4}{5}=0.8 \), so we can rewrite the equation as \( 7.2 = 0.8b \). To solve for \( b \), we divide both sides by \( 0.8 \). So \( b=\frac{7.2}{0.8} \). Calculating that, \( 7.2\div0.8 = 9 \). Let's check with the fraction method. If \( 7.2=\frac{4}{5}b \), we can multiply both sides by \( \frac{5}{4} \) (the reciprocal of \( \frac{4}{5} \)) to solve for \( b \). So \( b = 7.2\times\frac{5}{4} \). \( 7.2\times5 = 36 \), and \( 36\div4 = 9 \). So \( b = 9 \).

Looking at the boxes in the problem (even though the formatting is a bit unclear), let's map the steps:

1. First, when we distribute \( \frac{4}{5}(b - 5) \), we get \( \frac{4}{5}b-4 \), so the first box (after the \( \frac{4}{5}\cdot \)) should be \( b \), and the second box (after the \( \frac{4}{5}\cdot \)) should be \( 5 \) (since \( \frac{4}{5}(b - 5)=\frac{4}{5}b-\frac{4}{5}\times5 \)).

2. Then, the equation is \( 3.2=\frac{4}{5}b - 4 \). Then we add 4 to both sides: \( 3.2 + 4=\frac{4}{5}b \). So the next box (where it says \( 3.2=\square - 4 \)) should be \( \frac{4}{5}b \) (or in decimal, \( 0.8b \), but since we started with fractions, maybe \( \frac{4}{5}b \)). Then, when we add 4 to both sides, \( 3.2+4 = 7.2 \), so the box for \( 3.2+\square=\square \) would have 4 in the first square and \( \frac{4}{5}b \) (or 7.2? Wait, maybe the steps are:

Wait, let's re - examine the problem's boxes. The first line: \( 3.2=\frac{4}{5}\cdot\square+\frac{4}{5}\cdot\square \) (wait, no, the original is \( 3.2=\frac{4}{5}(b - 5) \), so distributing gives \( 3.2=\frac{4}{5}b-\frac{4}{5}\times5 \), so \( 3.2=\frac{4}{5}b - 4 \). Then, to solve, we add 4 to both sides: \( 3.2 + 4=\frac{4}{5}b \), so \( 7.2=\frac{4}{5}b \). Then, multiply both sides by \( \frac{5}{4} \): \( b = 7.2\times\frac{5}{4}=9 \).

So filling in the boxes:

- In the \( \frac{4}{5}\cdot\square+\frac{4}{5}\cdot\square \), the first square is \( b \) and the second is \( - 5 \)? Wait, no, the distributive property is \( a(b + c)=ab+ac \), but here it's \( a(b - c)=ab - ac \), so \( \frac{4}{5}(b - 5)=\frac{4}{5}b+\frac{4}{5}\times(- 5)=\frac{4}{5}b-4 \). So maybe the first box is \( b \) and the second is \( - 5 \), but that might be confusing. Alternatively, if we rewrite the equation as \( 3.2=\frac{4}{5}b-4 \), then we can add 4 to both sides: \( 3.2 + 4=\frac{4}{5}b \), so \( 7.2=\frac{4}{5}b \). Then, to solve for \( b \), we can do \( b=7.2\div\frac{4}{5}=7.2\times\frac{5}{4}=9 \).

So the boxes:

1. \( 3.2=\frac{4}{5}\cdot\boldsymbol{b}+\frac{4}{5}\cdot\boldsymbol{(- 5)} \) (but since \( \frac{4}{5}\times(-5)=-4 \), this is the distributive step)

2. \( 3.2=\boldsymbol{\frac{4}{5}b}-4 \)

3. \( 3.2+\boldsymbol{4}=\boldsymbol{\frac{4}{5}b} \) (or \( 3.2 + 4=7.2 \) and \( 7.2=\frac{4}{5}b \))

4. Then, \( \frac{5}{4}\times7.2=\boldsymbol{b} \), and \( \frac{5}{4}\times7.2 = 9 \), so \( b = 9 \)

Answer:

The value of \( b \) is \( \boldsymbol{9} \). And filling in the boxes (from left to right, top to bottom as per the problem's layout):

• In the \( \frac{4}{5}\cdot\square \) (first) : \( b \)

• In the \( \frac{4}{5}\cdot\square \) (second) : \( - 5 \) (or 5 with a negative sign, but since \( \frac{4}{5}(b - 5)=\frac{4}{5}b-\frac{4}{5}\times5 \), it's equivalent to \( \frac{4}{5}b+(-4) \))

• In the \( 3.2=\square - 4 \) : \( \frac{4}{5}b \) (or \( 0.8b \))

• In the \( 3.2+\square=\square \) : First square \( 4 \), second square \( \frac{4}{5}b \) (or \( 7.2 \))

• In the \( \frac{5}{4}\cdot\square=\square \) : First square \( 7.2 \), second square \( b \)

• And finally, \( b = 9 \)