All categories

3 years ago

How do i turn reapeating decimals into fractions?

1 answer:

6 0

Easy example that shows $0.999\ldots=1$ :

Let $x=0.999\ldots=0.\overline9$ .

Then $10x=9.999\ldots=9.\overline9$ .

So $10x-x=9.\overline9-0.\overline9$ , or $9x=9$ , and so $x=1$ .

The basic idea is to find the period of the repeating decimal, move the $n$ digits belonging to one period over to the left of the decimal point by multiplying by $10^n$ , then subtract the original repeating decimal from this new number, and finally divide by $10^n-1$ .

A slightly more complicated example:

Let $x=0.142857142857142857\ldots=0.\overline{142857}$ .

Then $10^6x=142857.\overline{142857}$ .

Then $10^6x-x=142857.\overline{142857}-0.\overline{142857}$ , or

$999999x=142857\implies x=\dfrac{142857}{999999}=\dfrac17$

You might be interested in

What is the volume of this rectangular prism if the side of each square is 1 cm?

aliina [53]

Your answer is B because it’s 6 times 3 times 4 and you add cubic because it’s volume.

7 0

2 years ago

Write a polynomial function $f$ of least degree that has rational coefficients, a leading coefficient of 1, and the given zeros.

olganol [36]

Answer:

$f(x) = x^{3} + 4x^{2} -7x^{} -10^{}$

Step-by-step explanation:

The zeros of the polynomial function are given us as -5,-1,2

If the zeros of a polynomial function are α,β,ω, the polynomial function can be obtained using the expression below:

f(x) = (x - α)(x - β)(x - ω)

where α = -5, β = -1, and ω = 2

$f(x)=(x-(-5) )(x -(-1))(x - 2) = (x+5)(x+1)(x-2)\\\\f(x)=(x+5)(x^{2} -2x + x -2) = (x+5)(x^{2} -x-2)\\\\f(x)=x(x^{2} -x-2)+5(x^{2} -x-2)\\\\f(x)= x^{3} - x^{2} -2x + 5x^{2} - 5x - 10\\\\f(x)= x^{3} + 4x^{2} -7x - 10$

<em>NB: To arrive at the answer, expand the brackets and after expansion, collect like terms to obtain the final answer</em>

4 0

3 years ago

One satellite is scheduled to be launched from Cape Canaveral in Florida, and another launching is scheduled for Vandenberg Air

Lisa [10]

Answer:

$P(A) = 0.45$

$P(B) = 0.32$

Step-by-step explanation:

Given

$P(A) > P(B)$

$P(A\ u\ B) = 0.626$

$P(A\ n\ B) = 0.144$

Required

Find P(A) and P(B)

We have that:

$P(A\ u\ B) = P(A) + P(B) - P(A\ n\ B)$ --- (1)

and

$P(A\ n\ B) = P(A) * P(B)$ --- (2)

The equations become:

$P(A\ u\ B) = P(A) + P(B) - P(A\ n\ B)$ --- (1)

$0.626 = P(A) + P(B) - 0.144$

Collect like terms

$P(A) + P(B) = 0.626 + 0.144$

$P(A) + P(B) = 0.770$

Make P(A) the subject

$P(A) = 0.770 - P(B)$

$P(A\ n\ B) = P(A) * P(B)$ --- (2)

$0.144 = P(A) * P(B)$

$P(A) * P(B) = 0.144$

Substitute: $P(A) = 0.770 - P(B)$

$[0.770 - P(B)] * P(B) = 0.144$

Open bracket

$0.770P(B) - P(B)^2 = 0.144$

Represent P(B) with x

$0.770x - x^2 = 0.144$

Rewrite as:

$x^2 - 0.770x + 0.144 = 0$

Expand

$x^2 - 0.45x - 0.32x + 0.144 = 0$

Factorize:

$x[x - 0.45] - 0.32[x - 0.45]= 0$

Factor out x - 0.45

$[x - 0.32][x - 0.45]= 0$

Split

$x - 0.32= 0 \ or\ x - 0.45= 0$

Solve for x

$x = 0.32\ or\ x = 0.45$

Recall that:

$P(B) = x$

So, we have:

$P(B) = 0.32 \ or \ P(B) = 0.45$

Recall that:

$P(A) = 0.770 - P(B)$

So, we have:

$P(A) = 0.770 - 0.32 \ or\ P(A) =0.770 - 0.45$

$P(A) = 0.45 \ or\ P(A) =0.32$

Since:

$P(A) > P(B)$

Then:

$P(A) = 0.45$

$P(B) = 0.32$

6 0

3 years ago

Helpers are needed to prepare for the fete. Each helper can make either 2 large cakes per hour, or 35 small cakes per hour. The

amid [387]

If 20 large cakes are needed, you'll need 4 helpers. To find this, you first need to find how many cakes 1 helper can make. Every hour, one helper can make 2 large cakes. If they have the kitchen for 3 hours, 6 large cakes can be made by one helper. This also means that 2 helpers make 12 cakes, 3 helpers can make 18 cakes, and 4 helpers can make 24 cakes. Since you can't have a fraction of a helper, 4 is your answer. For the 700 small cakes, you do the same thing. Every hour, 1 helper makes 35 small cakes. In 3 hours, 1 helper can make 105 small cakes. 2 can make 210, 3 can make 315, 4 can make 420, 5 can make 525, 6 can make 630, and 7 can make 735. The 700 small cakes takes 7 helpers and the 20 large cakes takes 4 helpers. Your total amount of helpers is 11.

8 0

3 years ago

What is 8 21/40 as a fraction or mixed number as a decimal

zepelin [54]

$8\frac{21}{40}=\frac{8\times40+21}{40}=\frac{341}{40}\\\\8\frac{21}{40}=8\frac{21\times25}{40\times25}=8\frac{525}{1000}=8.525$

8 0

3 years ago