Ask Uncle Colin: These percentages don’t add up!

Ask Uncle Colin is a chance to ask your burning, possibly embarrassing, maths questions -- and to show off your skills at coming up with clever acronyms. Send your questions to colin@flyingcoloursmaths.co.uk and Uncle Colin will do what he can.

Dear Uncle Colin,

I’m a bit confused! Someone told me that 40% of all Americans on welfare are black, which means that 66% of black Americans are on welfare. How did they work this out?

I don’t get it, and it’s bothering me.

— Always Back Equality Legistlation

Hi, ABEL! No wonder you’re confused, that 66% figure you’ve been given is way off.

I suspect your initial confusion comes from the problem of different sample spaces: the percentage of Americans receiving welfare who are black isn’t linked (at least not in a trivial way) to the percentage of black Americans receiving welfare.

The source doesn’t give the number of black Americans receiving welfare, but it’s easy enough to work out: it’s 39.8% (the percentage of recipients who are black) of 11,400,000 (the total number of Americans receiving welfare), which is 4,537,200 — given the inexactitude of the figures, I’ll round that to 4.5 million.

According to Wikipedia, there are around 37.7 million people of “non-Hispanic Black or African American” ethnicity in the USA, which makes the percentage of black Americans receiving welfare roughly $\frac{4,500,000}{37,700,000} \times 100 = 11.9%$ — again, accounting for roughness, somewhere around 12%, but certainly nowhere even close to 66%.

— Uncle Colin

PS: To quote Frank Zappa, I’m not black, but there are a whole lot of times I wish I could say I wasn’t white. This response isn’t about the widespread racial injustice in the USA, it’s not about discrimination, and it’s probably full of clumsy language that could be phrased in a better way. For all of these things, I apologise; my forte is writing about maths. Meanwhile, please follow Wheaton’s Law in the comments. I’m happy to correct errors of phrasing, but I will not tolerate racism or rudeness.

Colin

Colin is a Weymouth maths tutor, author of several Maths For Dummies books and A-level maths guides. He started Flying Colours Maths in 2008. He lives with an espresso pot and nothing to prove.

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