MathProblemSolver101

Where's the equation?

James used 4/7 of his screws and 3/5 of his nails to construct a wooden cupboard. In the end, he had an equal number of screws and nails left. Given that James had a total of 145 screws and nails at first, how many nails did he use to construct the wooden cupboard?

Let s = the number of screws

Let n = the number of nails

We have: $(1-\frac{4}{7})s = (1-\frac{3}{5})n$

This is equal to $\frac{3}{7}s = \frac{2}{5}n.$

We also have $n+s = 145.$

If we multiply the first equation by its common denominator namely $7 \cdot 5 = 35,$ we have that $15s = 14n.$

Multiplying the second equation by 15 we have $15n + 15s = 145 \cdot 15.$

Let’s substitute $14n$ into $15s$: $15n + 14n = 145 \cdot 15 \Rightarrow 29n = 145 \cdot 15.$

Divide both sides by $29:$ $n = 5 \cdot 15 = \boxed{75}$

 - Jimmy

On the day after Halloween, Kabir had 5 times as much candy as Sam. If Kabir gives 18 pieces of candy to Sam, they will each have the same amount. How many pieces of candy do they have in all?

Let k = number of candies Kabir has.

Let s = number of candies Sam has.

We have $k = 5s \qquad$ (1)

We also have $k-18=s+18 \qquad$ (2)

(1) * (2) => $5s - 18 = s + 18$

$4s = 36$

$s = 9 \qquad$ (3)

(3) & (1) => $k = 45 \qquad$ (4)

(3) + (4) => $9 + 45 = \boxed{54}$

By definition, $r = \frac{d}{2}.$

That is a very hard question!

$2x^2 - 7x + 1 = 8x^2 - 25x + 13$

$6x^2 - 18x + 12 = 0$

$x^2 - 3x + 2 = 0$

$(x-2)(x-1)=0$

$x=2; x=1$

$\boxed{(2,0), (1,0)}$

$\frac{3-(-2)}{-4-(6)}=\frac{5}{-10}=-\frac{5}{10}=\boxed{-\frac{1}{2}}.$

Since there are only two ways to end to $B$, we have 2 cases.

Case 1: It ends with $\overline{FB}.$

There are 6 paths: $A \Rightarrow C \Rightarrow F \Rightarrow B$

$A \Rightarrow C \Rightarrow D \Rightarrow E \Rightarrow F \Rightarrow B$

$A \Rightarrow D \Rightarrow C \Rightarrow F \Rightarrow B$

$A \Rightarrow D \Rightarrow E \Rightarrow F \Rightarrow B$

$A \Rightarrow D \Rightarrow F \Rightarrow B$

$A \Rightarrow C \Rightarrow D \Rightarrow F \Rightarrow B.$

Case 2: It ends with $\overline{CB}.$

There are 4 paths: $A \Rightarrow D \Rightarrow C \Rightarrow B$

$A \Rightarrow D \Rightarrow F \Rightarrow C \Rightarrow B$

$A \Rightarrow C \Rightarrow B$

$A \Rightarrow D \Rightarrow E \Rightarrow F \Rightarrow B$

Adding these cases yields $6+4=\boxed{10 \text{ ways.}}$

If anyone finds a better way, please hit me up as this casework a bit tedious to do.

 - Jimmy

A line has a slope of 8/7 and its y-intercept is (0,18). What is its x-intercept?

$y = mx + b$

$y = \frac{8}{7}x + 18$

When $y=0,$ the result is the x-intercept.

$0 = \frac{8}{7}x + 18$

$0 = \frac{4}{7}x + 9$

$0 = 4x + 63$

$4x = -63$

$x = - \frac{63}{4} = -15 \frac{3}{4}$

The x-intercept is $\boxed{(-15 \frac{3}{4}, 0)}$

$2^{\frac{4}{12}}=2^{\frac{1}{3}}=\sqrt[3]{2^1}=\boxed{\sqrt[3]{2}}$