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