What are some concrete examples from the text that show, despite the government's extreme efforts, people still aren't equal in this society?

The premise of Vonnegut's short story "Harrison Bergeron" is society cannot be successful when trying to create equality by stripping individuality from its people. 

In the short story the government tries to give people who are above average intelligence, strength, or beauty "handicaps" in order to give the illusion of equality. For example, Harrison's father George is given a "little mental handicap radio in his ear" because "his intelligence was way above normal." This radio transmits a "sharp noise to keep people like George from taking unfair advantage of their brains." People must not remove their handicaps, or they will be punished. 

Despite the government's efforts, it is evident the system of handicapping individuals does not work. A prime example of this is the dynamic between George and Hazel (Harrison's parents). While watching television, Hazel is aware that George's transmitter is emitting a sound as she watches him "wince." Although she does not have to wear a handicap, she deals with the emotional burden of watching her husband suffer from having his. She wants her husband to have some respite from the pain the transmitter inflicts on him. She tells him to rest because she "[doesn't] care if [George is] not equal to [her] for a while.” This dialogue indicates that the government's attempts to make people behave and appear as equals does not have an impact on their interactions with one another. Basically, the handicaps are not causing the effect that the government hopes because people still recognize that they are not equal. 

This is further illustrated through the narration of the story. The narrator describes the characters in relation to their handicaps. By doing so, this focuses the reader's attention to the handicaps and what they mask, rather than how the characters are now equal as a result of the handicaps. An example of this is when George and Hazel watch the ballet. The narrator describes the dancers in terms of their handicaps and how the handicaps reveal how talented or beautiful they really are. As they dance, the ballerinas "were burdened with sash weights and bags of birdshot, and their faces were masked, so that no one, seeing a free and graceful gesture or a pretty face." Rather than detracting from their beauty and talent, the handicaps emphasize and draw attention to their above average qualities. 

An electrician uses tiny red, blue and yellow bulbs in a ratio 12:11:10 to light up an entire building. If 4752 bulbs were used in all, how many...

If we have 12+11+10 = 33 bulbs then;

12/33 is red

11/33 is blue

10/33 is yellow

Similarly if we have 4752 bulbs;

Number of red bulbs `= (12/33)xx4752 = 1728`

Number of blue bulbs  `=(11/33)xx4752 = 1584`

Number of yellow bulbs `=(10/33)xx4752 = 1440`

Bud has a very active imagination. How does this help him throughout the book? Does it ever get him in trouble? Why do you think imagination is so...

Bud's active imagination gives him hope throughout the novel and encourages him to continue his journey during difficult times. Bud compares his thoughts to tiny seeds that grow into massive maple trees. Bud's initial thoughts about his father develop into an obsession to find him. The thought of living a carefree life with his father motivates Bud to travel across Michigan to find him. Bud's imagination also soothes him in unfamiliar environments. Memories of Bud's mother calm him down and make him comfortable no matter where he is at.

Sometimes, Bud's imagination gets him into trouble. At the beginning of the novel when Bud is locked inside the Amos' shed, Bud thinks that there is a vampire bat hanging from the ceiling. Bud fears that the vampire bat will suck his blood, so Bud picks up a rake, and swings at it as hard as he can. The "vampire bat" turns out to be a hornet's nest and Bud gets stung badly. Another moment where Bud's imagination gets him into trouble is when Lefty Lewis offers him a ride. Bud sees an icebox labeled "URGENT: CONTAINS HUMAN BLOOD," and thinks that Lefty must be a vampire. Bud locks Lefty out of the car and attempts to drive away. Fortunately, Bud does not make it very far before the car shuts off.

Having an active imagination would benefit individuals who lived during the Depression. The Depression, aptly named, was a rough period to be alive. In order for people to have hope and maintain a positive outlook on life, they would need to have an active imagination to counter the dismal realities of the world around them.

There have been numerous difficult moments throughout my life where my imagination has helped distract my attention and relieve my stress. From imagining a picturesque beach when its cold and rainy outside, to thinking about playing catch with my father to soothe my mind before taking an important test, my imagination helps me through difficult situations each day.

What was one question that miss Thomas and the band members had never asked Bud?

Miss Thomas, Herman E. Calloway, and the band members had never asked Bud what his mother's name was.

One might wonder how they could possibly not think of asking such an obvious question. When Bud showed up claiming to be Mr. Calloway's son, he hinted that his "Momma" told him so, but "not in the words just like that." When miss Thomas pressed him, he did not volunteer the information about the band's flyers his mother had saved, so the issue was dropped. Interestingly, the band members believed Bud when he said that his "Momma" died six years ago at the age of twenty-six, and Miss Thomas believed, for the most part, his story about the orphanage and the foster family. However, the idea that Mr. Calloway might be his father, or somebody Bud's mother might have known, seemed so far-fetched and ludicrous that it crossed no one's mind to inquire further why and how Bud came up with it. Only when Bud showed Mr. Calloway the rocks with inscriptions of the places and dates, similar to the ones in his car, was the investigation of Bud's claim renewed, finally leading to the discovery of his mother's identity.

Please Explain this stanza: Soul lies buried in conscience / Body is fool's ornament / Man behind a fatal fence / Around which he never went....

Without knowing the poem this stanza is from, it's not possible to explain it in its broader context, but these four lines hold together as one thought and can be explained in terms of a Freudian concept. The "soul" is the innate spirit of a person who has natural inner wisdom. This inner wisdom is suppressed by the socio-cultural "conscience" superimposed upon the soul by the concept known as the Freudian Super-Ego. The "body" is represented as separate from and inferior to the "soul" and is additionally represented as the "fatal fence" that entraps the "soul": the body is the untamed being that requires the conscience developed by the Super-Ego and that buries the soul, "The Soul lies buried...." Resulting in the entrapment of the soul, the person ("Man") never ventures to go out around the body: the person never dares to transcend the physical and soar in transcendental spirituality to find birth and life instead of burial; freedom from the conscience, the body, the "fatal fence" and the "fool's ornament" that bind the soul in metaphysical death.

Which characters were most concerned about their reputations in Arthur Miller's, The Crucible?

The first and most obvious, is the Reverend Samuel Parris, the village priest. Reverend Parris used to be a merchant and had a history of inviting derision wherever he went. When he arrived in Salem him brought with him not only his slave, Tituba, from Barbados, his daughter, Betty and niece, Abigail Williams, but also a long history of paranoia.

It did not take the Reverend long to feel that he was being persecuted by members of his congregation. He was terrified about being ousted from his position and had a need to protect his reputation at all costs. He felt that there was a 'faction' led specifically by John Proctor, Rebecca Nurse and Giles Corey whose sole purpose was to rid Salem of him, so when rumours of witchcraft were traced to his household because of the actions of his slave, daughter and niece, he made every attempt to clear his name.

The Reverend was so anxious about this that he constantly meddled in the court's affairs during the witch trials, to such an extent that he irritated Judge Danforth who, out of contempt for him, commanded him to shut up at one point. The Reverend's desperate attempts to retain his reputation and his position were, however, an exercise in futility. When the trials ended he was voted out of his post and left Salem, never to be seen or heard of again.

Another villager who was keen to protect his reputation was Thomas Putnam, the son of the wealthiest resident in the village. He had many grievances and huge resentment towards many in the village. He had consistently tried to manipulate affairs in the village to improve his reputation and status, with limited success. His failures in this regard made him a deeply embittered man, for, as Arthur Miller states: 

... he regarded himself as the intellectual superior of most of the people around him.

and

Thomas Putnam felt that his own name and the honor of his family had been smirched by the village, and he meant to right matters however he could.

It is primarily for these reasons and his greed for others property that we find Thomas Putnam's names on most of the indictments against others for witchcraft. The trials offered him an opportunity to, as he believed, vindicate his good name and also, to enrich himself and gain greater status and authority.

Another individual intent on, not only to enhance his reputation but to assert it in whichever way he possibly could, was the mean and unscrupulously arrogant and stubborn head of proceedings during the trials, Deputy Governor Danforth. The judge was much aware of his reputation and consistently reminded witnesses and accused of his status and power, lest they forget. He, for example asks Francis Nurse the following during trial:

And do you know that near to four hundred are in the jails from Marblehead to Lynn, and upon my signature?

And seventy-two condemned to hang by that signature?

The judge obviously relishes his reputation as a no-nonsense dispenser of justice. No one dare challenge his authority, for they will face his most severe retribution. He is a commanding and dictatorial force throughout the trial, stamping his personality, authority and reputation on all the proceedings.

Finally, one has to mention the ill-fated John Proctor, our chief protagonist. His reputation was important to him for all the right reasons. John took a strong moral stance in defending his name once he had given his persecutors what they wanted: a false confession. He even went as far as signing such confession but when asked to hand it over, he balked. He would not have his name tarnished any further. His confession would be displayed for everyone to see. Since he was a respected man in the village, it would encourage others to follow his example. When asked why he refuses to hand over his confession, John in agony for what he had done, passionately cries out:

Because it is my name! Because I cannot have another in my life! Because I lie and sign myself to lies! Because I am not worth the dust on the feet of them that hang! How may I live without my name? I have given you my soul; leave me my name!

And in his final words before he is led to the scaffold, he tells Elizabeth:

Give them no tear! Tears pleasure them! Show honor now, show a stony heart and sink them with it!

How does the author use Wolf's apprehension to foreshadow what is to come in the story?

Irving first establishes a deep connection between his sketch's eponymous protagonist and his dog, Wolf, in order to set the stage for his eventual apprehension:

Rip’s sole domestic adherent was his dog Wolf, who was as much hen-pecked as his master; for Dame Van Winkle regarded them as companions in idleness, and even looked upon Wolf with an evil eye, as the cause of his master’s going so often astray.

Describing Wolf as "Rip's sole domestic adherent" highlights their relationship vis-a-vis "Dame Van Winkle," who thinks that both of them are lazy "companions in idleness." Irving underscores this loyalty by telling the reader that Wolf "was as courageous an animal as ever scoured the woods."

This connection elevates the significance of his initial apprehension regarding the strange people playing in the mountain glen. Upon hearing his name called, Rip begins to approach the men, and

at the same time Wolf bristled up his back, and giving a low growl, skulked to his master’s side, looking fearfully down into the glen. Rip now felt a vague apprehension stealing over him; he looked anxiously in the same direction, and perceived a strange figure slowly toiling up the rocks, and bending under the weight of something he carried on his back.

Because Rip and Wolf are so intertwined in the narration, this "bristl[ing] up his back, and giving a low growl" on the dog's part implies that there is danger ahead for both he and his master. This is why Rip feels "a vague apprehension," which foreshadows the decades-long slumber that is about to befall him after drinking with the strange men in the glen.