Master Fraction Math Problems with Expert Help

Find the prime factorisation of $980$ .

Gharib4Pe 2022-11-24

Which of the following are equivalent fractions of $\frac{2}{3}$ ?

vegetatzz8s 2022-11-24

What is the reciprocal of $\frac{- 2}{3}$ ?

Dayanara Nash 2022-11-24

Convert $0.35$ to a fraction in lowest terms

Ernesto Wagner 2022-11-24

$66 \frac{2}{3} %$ as a fraction is $\underline{}$ .
A $\frac{2}{3}$
B $\frac{1}{4}$
C $\frac{1}{6}$
D $\frac{1}{3}$

siotaody 2022-11-24

How to prove $\sum_{i = 1}^{n - 1} \frac{1}{lcm (a_{i}, a_{i + 1})} < 1$ where $a_{i} \in N$ and $a_{i} < a_{i + 1}$ ?

Cecilia Wilson 2022-11-24

If $A$ is a real diagonalizable matrix and $B$ is real symmetric and positive definite, is the matrix $B^{- 1} A$ real diagonalizable?
Let G be the square root of B and $G = G^{T}$ , then $B^{- 1} A = G^{- 1} (G^{- 1} A G^{- T}) G$ is similar to $G^{- 1} A G^{- T}$ , which is again symmetric and thus diagonalizable.
However, this method fails when A is just real diagonalizable. Are there any tricks to modify the proof, or is there a counterexample?

umthumaL3e 2022-11-23

What is $2.25$ as a fraction?

Frankie Burnett 2022-11-21

Any ideas for this cyclic inequality?
If $a, b, c, d > 0$ and $a + b + c + d = 8$ , prove that
$\frac{a}{b^{2} + 2 b} + \frac{b}{c^{2} + 2 c} + \frac{c}{d^{2} + 2 d} + \frac{d}{a^{2} + 2 a} ⩾ \frac{16}{(a + c) (b + d)} .$

Leanna Jennings 2022-11-21

$- 2 \frac{3}{5}$ " mean?
I have seen this notation before: $2 \frac{3}{5}$ . It means $2 + \frac{3}{5}$ I know it since $3$ rd grade. It always seemed confusing to me because sometimes (especially in physics problems), $2 \frac{3}{5}$ might mean $2 \cdot \frac{3}{5}$ . But it math, I know it means addition. The problem is that I have never used this notation before with negative integers like in $- 2 \frac{3}{5}$ . Logically, it means negative $2$ and $3 / 5$ , which would be $- (2 + \frac{3}{5}) = - 2 - \frac{3}{5}$ . However, shouldn't this be noted as $- (2 \frac{3}{5})$ ? What does $- 2 \frac{3}{5}$ really mean?

pighead73283r 2022-11-20

Is it possible to manipulate: $\frac{x^{2} - 2}{x + 1}$ such that the numerator has a lower power than the denominator?
Is it possible to manipulate: $\frac{x^{2} - 2}{x + 1}$ such that the numerator has a lower power than the denominator?
For example: $\frac{x + 3}{x + 2}$ would be $1 + \frac{1}{x + 2}$

Audrey Arnold 2022-11-20

Does a fraction that can be simplified as an integer belong to the set of integer numbers?
Consider the question:
$\frac{12}{2} \in Z$
Would you consider the above statement mathematically correct? and why?
I'm a bit on the side of calling it a true statement. However, since $\frac{12}{2}$ is equivalent to 6 but not exactly identical, I am quite skeptical.
Also, how about the case:
$\frac{12}{1} \in Z$
I checked this question Are all integers fractions?. it has some insight addressing a similar issue, yet some answers make it even more confusing for me.
Thank you!

kunguwaat81 2022-11-20

If $x + y + z + w = 5$ then the minimum value of $x^{2} \cot (9 °) + y^{2} \cot (27 °) + z^{2} \cot (63 °) + w^{2} \cot (81 °)$ is?
I used the $A . M . \geq G . M .$ rule.
$((x y z w)^{2} . \cot (9^{o}) \tan (9^{o}) \cot (27^{o}) \tan (27^{o}))^{1 / 4} \leq \frac{x^{2} \cot (9 °) + y^{2} \cot (27 °) + z^{2} \cot (63 °) + w^{2} \cot (81 °)}{4}$
This reduces to :
$4 (x y z w)^{1 / 2} \leq x^{2} \cot (9 °) + y^{2} \cot (27 °) + z^{2} \cot (63 °) + w^{2} \cot (81 °) - - - - (1)$
Now, I applied the same thing to the first equation:
$\frac{x + y + z + w}{4} \geq (x y w z)^{1 / 4}$
$\frac{5}{4} \geq (x y w z)^{1 / 4}$
$\frac{25}{16} \geq (x y w z)^{1 / 2} - - - - (2)$
Now I do not know what to do. From equation (2), I get the minimum value of $(x y w z)^{1 / 2}$ as $- \infty$ (Am I right over here ?). In a nutshell, I am confused what to do next.

Annie French 2022-11-20

Given $a + b + c + d = 4$ To find Minimum value of $\frac{a}{b^{3} + 4} + \frac{b}{c^{3} + 4} + \frac{c}{d^{3} + 4} + \frac{d}{a^{3} + 4}$
Given $a + b + c + d = 4$ where $a, b, c, d \in R^{+}$
find Minimum value of
$S = \frac{a}{b^{3} + 4} + \frac{b}{c^{3} + 4} + \frac{c}{d^{3} + 4} + \frac{d}{a^{3} + 4}$
I have no clue to start...any hint?

mxty42ued 2022-11-20

Is a decimal number, such as 0.345 a fraction?
Just by semantics alone, would you consider it a fraction when it is not written in a 'fractional' format?

Kayden Mills 2022-11-19

How do I equate coefficients?
I got the problem into this form:
$A + B = 0$
$C - B = 1$
$8 A + 4 B - C + D = 10$
$- 4 B + 4 C - D + E = 3$
$16 A - 4 C - E = 36$
I really don't know what to do after that. I suck at equating coefficients. It never makes sense to me. Please help and provide steps and a good explanation. Thank you.

Elliana Molina 2022-11-18

What is the mixed number of $\frac{105}{13}$

Kale Sampson 2022-11-18

Why is a machine producing 7 nails in 9 seconds faster than one producing 11 nails in 14 seconds?
Machine A Produces 11 nails in 14 seconds.
Machine B produces 7 nails in 9 seconds.
Which is faster?
I thought that $\frac{11}{14}$ is larger than $\frac{7}{9}$ and thus A is the answer. However, the answer says that $\frac{9}{7}$ is bigger than $\frac{14}{11}$ so B is the answer. How does this work?

bruinhemd3ji 2022-11-17

Proof of an $\frac{a}{b} = \frac{c}{d} ⟺ \frac{a + b}{a - b} = \frac{c + d}{c - d}$
but going from $\frac{a}{b} = \frac{c}{d}$
I already proved it backwards
$\frac{a + b}{a - b} = \frac{c + d}{c - d}$ I multiply each side by (a-b)(c-d)
$a c - a d + b c - b d = a c - b c + a d - b d$ so $2 b c = 2 a d$ which by dividing both sides by $2 b d$ gives us :
$\frac{a}{b} = \frac{c}{d}$
any ideas how to proceed ?
Edit : this equivalence only works when $a \neq b$ and $c \neq d$

Emma Hobbs 2022-11-17

$33 \frac{1}{3}$ percent of 600

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