Ralph suggested that they needed to build a fire to attract any passing planes or ships if they wanted to be rescued. Jack led the boys up to to the top of the mountain to build the fire. Both Ralph and Jack worked hard to build a huge bonfire and then, they realised that they had nothing to light it with. They had no matches, no cigarette lighter, nothing. That is why they were both embarrassed.
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Jack had focused on the hunting aspect of survival, while Ralph concentrated on building shelter so that they could be safe from nature. So later on, Jack went to hunt a pig while Simon and Ralph were busy building huts. Ralph and Jack argue over the importance and priorities of what had to be done.
Both are capable leaders who are able to command support and respect but employ completely different leadership styles.
Ralph and Jack never liked each other. Once they met and Jack wanted to take the power from Ralph there was instant friction between the two. It's was obviously Ralph's job to be chief and Jack just expected to get it instead of Ralph.
Ralph interrupts and bitterly accuses them of irresponsibility for letting the fire burn out. The hunters, still in the grip of the thrill of the hunt, ignore him. Ralph once again tries to make them realize the enormity of their blunder. In quick and angry retaliation, Jack strikes out at Piggy, forcefully slapping his face and breaking one lens of his glasses
Ralph told Samneric that he would hide out in the bushes, so eventually Samneric squeal on Ralph to Jack.
Jack challenged Ralph to go up the mountain to look for the beast with him in the dark. Ralph responds by accepting the challenge and bringing Roger along with him.
Ralph agreed to go with Jack because he was showing Jack he wasn't scared to kill the beast. Jack taunted Ralph and so Jack didn't feel like he had won he agreed to go with ease.