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Three babies are playing with a set of building blocks.
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Alfie has some number of blocks.
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Bertie has four times as many blocks as Alfie.
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Cara has three less blocks than Bertie.
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The babies have a total of 51 blocks.
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One of the parents says, “If Cara gives some blocks to Alfie, then Cara and Alfie will have the same number of blocks.”
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Is this statement correct?
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You must show your reasoning.
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Let’s start by defining the number of blocks that Alfie has.
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Since we don’t know exactly how many blocks that is, we can give it a letter.
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Let’s call it 𝑎, where 𝑎 is an integer or a whole number.
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Remember since 𝑎 is the number of blocks that Alfie has, it simply has to be a whole number since we can’t split these blocks into pieces.
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This means then that Bertie has four 𝑎 blocks since four 𝑎 means four times 𝑎 and we’re told that he has four times as many blocks as Alfie.
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We’re told that Cara has three less blocks than Bertie.
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And we’ve just stated that Bertie has four 𝑎 blocks.
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So Cara must have four 𝑎 minus three blocks.
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We’re also told that the total number of blocks is 51.
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So we can add together each of our expressions and form an equation by making the total equal to 51.
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That’s 𝑎 plus four 𝑎 plus four 𝑎 minus three equals 51.
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Next, we should simplify by collecting like terms.
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𝑎 plus four 𝑎 plus four 𝑎 is nine 𝑎.
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So our equation is nine 𝑎 minus three equals 51.
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Next, we’ll solve this equation by applying the inverse operations.
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First, to get rid of the minus three, we’ll add three to both sides of the equation.
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That gives us nine 𝑎 is equal to 54.
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Nine 𝑎 means nine multiplied by 𝑎.
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So the inverse operation here is to divide both sides by nine.
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That gives us that 𝑎 is equal to six.
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That means Alfie has six blocks.
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Since Bertie has four times the number of blocks Alfie has, he must have four multiplied by six which is 24 blocks.
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Cara has three less than this and 24 minus three is 21.
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We can see then that Alfie has six blocks and Cara has 21.
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Is there a way then for these two children to share their blocks so that they have the exact same number?
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Well, first, if we add together the total number of blocks that Alfie and Cara have, we get 21 plus six is equal to 27.
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27 cannot be split between two people without splitting a block.
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So the parent is wrong.