Michael bought three muffins and two cookies for three dollars 30, while James bought two muffins and five cookies for five dollars 50. Work out the price of a single muffin and a single cookie.
If we let a single muffin equal 𝑥 and a single cookie equal 𝑦, we can set up two equations. Firstly, as Michael bought three muffins and two cookies, we can say that three 𝑥 plus two 𝑦 equals 330, as it cost him three dollars 30 or 330 cents.
In the same way, James bought two muffins and five cookies. And this cost him five dollars 50 or 550 cents. Therefore, two 𝑥 plus five 𝑦 equals 550. We now have a pair of simultaneous equations which we can solve in order to work out the value of 𝑥, a muffin, and 𝑦, a cookie.
If we multiply the top equation by two and the bottom equation by three, we can make the 𝑥 coefficients the same. Three 𝑥 multiplied by two is six 𝑥. Two 𝑦 multiplied by two is four 𝑦. And 330 multiplied by two is 660.
In the same way, two 𝑥 multiplied by three is six 𝑥. Five 𝑦 multiplied by three is 15 𝑦. And 550 multiplied by three is 1650. Subtracting equation one from equation two gives us 11 𝑦 equals 990.
If we divide both sides of this equation by 11, we are left with a value for 𝑦 equal to 90 cents. Therefore, a single cookie cost 90 cents. Substituting this value for 𝑦 into the top equation will allow us to calculate 𝑥.
Three 𝑥 plus two multiplied by 90 is equal to 330. As two multiplied by 90 is 180, we are left with three 𝑥 plus 180 equals 330. Subtracting 180 from both sides of this gives us three 𝑥 equals 150.
And finally, dividing by three gives us a value for 𝑥 of 50 cents. The muffin costs 50 cents. Therefore, our overall solution to the question is that a single muffin cost 50 cents and a single cookie cost 90 cents.