Question

ANSWER PROPERLY OR ELSE.
SJOW YOU WORKING OR ELSE.

Answer 1

see attached

Explanation:

You want the boxes of the pyramid shown filled with mixed numbers such that each box is the sum of the two numbers below it.

Strategy

As the example shows, each box is filled with the difference between the box above and the one adjacent. Work must necessarily proceed from the top down.

Math

As you know, the difference of fractions can be computed as ...

`(a)/(b)-(c)/(d)=(ad-bc)/(bd)`

Of course the number 1 can be written in terms of denominator d as ...

1 = d/d

This can be used to adjust the integer and fractional parts so both are positive.

We will use subscripts rc to indicate the row and column of a box, counting rows from the top down.

`X_(22)=12(3)/(4)-8(2)/(3)=(12-8)+\left((3)/(4)-(2)/(3)\right)=4(9-8)/(12)=\boxed{4(1)/(12)}\\\\\\X_(31)=8(2)/(3)-2(4)/(5)=(8-2)+\left((2)/(3)-(4)/(5)\right)=6(10-12)/(15)=\boxed{5(13)/(15)}\\\\\\X_(33)=4(1)/(12)-2(4)/(5)=(4-2)+\left((1)/(12)-(4)/(5)\right)=2(5-48)/(60)=\boxed{1(17)/(60)}`

`X_(41)=5(13)/(15)-1(5)/(9)=(5-1)+\left((13)/(15)-(5)/(9)\right)=4(39-25)/(45)=\boxed{4(14)/(45)}\\\\\\X_(43)=2(4)/(5)-1(5)/(9)=(2-1)+\left((4)/(5)-(5)/(9)\right)=1(36-25)/(45)=\boxed{1(11)/(45)}\\\\\\X_(44)=1(17)/(60)-1(11)/(45)=(1-1)+\left((17)/(60)-(11)/(45)\right)=(51-44)/(180)=\boxed{(7)/(180)}`

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Additional comment

In a couple of places the denominators have a common factor.

For example, 13/15 and 5/9 have denominators that have a common factor of 3: 15=3·5 and 9=3·3. When we did the cross multiplication to find the difference, we divided the cross products by that common factor so the resulting fraction would be in lowest terms.

That is, we used 13·9/3 = 39 and 15·5/3 = 25 for the numerator values, and 15·9/3 = 45 for the denominator value.

You will also recognize that 6+(-2/15) became 5+13/15 by using 6 = 5+15/15.

Many calculators and spreadsheets can perform arithmetic on mixed numbers and give a mixed number result. (That is how we checked our work.)