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Planning a budget for an event

Ratio, rates and money · level 3, lesson 6 of 6

A question to start

Your class of 24 has 60 dollars in its class fund for an end-of-year party. Everyone has ideas: pizza, juice, balloons, popcorn, a cake, music, prizes.

Here is the puzzle. If you just start buying things you like, you will run out of money before the party is ready. If you are too careful, you will have money left over and a dull party. How do you plan so that everyone gets fed, you have some fun extras, and you do not go over 60 dollars, not even by a cent?

Planning money like this has a name: making a budget. Every lesson in this level helps: unit rates, rounding, sharing in a ratio, and even mixing drinks.

The idea

What a budget is

A budget is a plan for money. It has two sides:

  • What you have: here, the 60 dollars. This is a fixed amount. It does not grow because you want more things.
  • What you will spend it on: a list of items with their costs.

A budget balances (or "works") when the total spent is less than or equal to what you have. The money not spent is left over (some people call it the surplus). If the total is more than you have, the budget is over, and something must change before you buy anything.

Needs first, then wants

In level 2 you sorted things into needs and wants. For a party, the needs are the things that make it a party at all: food, drink, plates. The wants are nice extras: balloons, popcorn, prizes.

Good budgeting puts the needs on the list first. Then you see how much is left, and choose wants that fit. If you start with the wants, you may find there is no money left for the food.

How many do you need? Use "for every"

Each item comes in packs, and you need enough for 24 people. This is ratio thinking.

  • A large pizza has 8 slices. With 1 slice for every child, you need 24 slices, and \(24 \div 8 = 3\) pizzas.
  • Juice boxes come in packs of 10. You need 24 boxes, and \(24 \div 10 = 2.4\) packs. You cannot buy 0.4 of a pack, so you must buy 3 packs (30 boxes, 6 spare).

When a division does not come out even, think about what makes sense. For things you must have enough of, round up. Buying 2 packs would leave 4 children with nothing to drink.

A budget table

Writing the plan as a table keeps everything clear. (In this lesson, prices include any tax. In level 4 you will add sales tax yourself.)

Item Price each How many Cost
Large pizza (8 slices) 12.99 3 38.97
Juice boxes, pack of 10 4.49 3 13.47
Plates and napkins 3.50 1 3.50
Total for the needs 55.94

Left over for wants: \(60.00 - 55.94 = 4.06\) dollars.

Now the wants. Balloons cost 2.99 dollars a pack and popcorn costs 2.49 dollars a bag. Both together cost \(2.99 + 2.49 = 5.48\) dollars, which is more than 4.06. So the class can choose one: balloons (leaving 1.07 dollars) or popcorn (leaving 1.57 dollars). That is a real choice, and a vote would settle it.

Estimate first: round up

Before adding exact prices, a quick estimate tells you if the plan is even close. For a budget, the safe way is to round each price up:

  • pizza: about 13 dollars, times 3 is 39;
  • juice: about 4.50 dollars, times 3 is 13.50;
  • plates: 3.50.

Estimate: \(39 + 13.50 + 3.50 = 56\) dollars.

Why round up? Because then your estimate is a little more than the real total. If the rounded-up total fits inside your budget, the real total must fit too. If you rounded down, the estimate might fit while the real cost does not, and you would get a nasty surprise at the till.

The cost per person

The cost per person is a unit rate: total cost divided by the number of people. For the needs above, \(55.94 \div 24 \approx 2.33\) dollars per child. Turning it round: a budget of 60 dollars for 24 children allows at most \(60 \div 24 = 2.50\) dollars per child. Comparing the two (2.33 is less than 2.50) is another quick way to see the plan fits.

When the plan does not fit

Plans change. Suppose 4 more children join the party, so there are 28. Now you need 28 slices: \(28 \div 8 = 3.5\), so 4 pizzas (rounding up).

Item Price each How many Cost
Large pizza 12.99 4 51.96
Juice boxes, pack of 10 4.49 3 13.47
Plates and napkins 3.50 1 3.50
Total 68.93

That is \(68.93 - 60.00 = 8.93\) dollars over. You have three kinds of fix:

  1. Spend less on an item: look for a better unit price, or a cheaper version.
  2. Cut something: drop an item, or have less of it.
  3. Get more money: a bake sale, or asking each family for a little.

You will see one way to fix it in worked example 3. There is rarely only one right answer; a budget is a set of choices, and math tells you which choices work.

Worked examples

1. The basic plan. This is the table above. Needs cost 55.94 dollars, leaving 4.06 dollars. Balloons (2.99) or popcorn (2.49) fit, but not both (5.48).

2. Estimating. A shopping list has items at 4.95, 7.20, 2.35 and 11.80 dollars. Will 30 dollars be enough?

  • Round each price up to a whole dollar: \(5 + 8 + 3 + 12 = 28\) dollars.
  • The rounded-up total is under 30, so the real total is too. Yes, 30 dollars is enough.
  • Exact total: \(4.95 + 7.20 + 2.35 + 11.80 = 26.30\) dollars.

3. Fixing the 28-child party. The class finds a deal: 2 large pizzas for 22 dollars. And instead of juice boxes, they make lemonade from frozen concentrate at 1.99 dollars a can. The label says to mix 1 can of concentrate with 4 cans of water, and in this example each can makes 8 cups. For 28 cups they need \(28 \div 8 = 3.5\), so 4 cans (rounding up again).

Item Price How many Cost
Pizza deal (2 large) 22.00 2 44.00
Lemonade concentrate 1.99 4 7.96
Plates and napkins 3.50 1 3.50
Total 55.46

Now the needs fit, with \(60.00 - 55.46 = 4.54\) dollars left: enough for balloons or popcorn, but not both (5.48).

4. Sharing a cost in a ratio. Two classes share a party that costs 85 dollars. Class A has 24 children and class B has 16. How should they split the cost fairly?

  • The ratio of children is \(24 : 16\). Both divide by 8: \(3 : 2\).
  • Boxes: \(3 + 2 = 5\). One box: \(85 \div 5 = 17\) dollars.
  • Class A pays \(3 \times 17 = 51\) dollars; class B pays \(2 \times 17 = 34\) dollars.
  • Check it is fair per child: \(51 \div 24 = 2.125\) and \(34 \div 16 = 2.125\). The same, about 2.13 dollars each.

5. Raising the money. The class wants 90 dollars for a trip and sells cookies at 1.50 dollars each. How many cookies must they sell?

  • \(90 \div 1.50 = 60\) cookies.
  • But the cookie ingredients cost 15 dollars. To end with 90 dollars after paying for them, they must take in \(90 + 15 = 105\) dollars, so \(105 \div 1.50 = 70\) cookies.

Try it

  1. You have 40 dollars and buy items costing 12.50, 9.75 and 8.20 dollars. How much is left?
  2. Estimate by rounding up: will 25 dollars cover items at 6.40, 3.15, 9.99 and 2.50 dollars? Then find the exact total.
  3. A large pizza (8 slices) costs 12.99 dollars. How many pizzas, and what cost, for 30 people having one slice each?
  4. Juice boxes come in packs of 10 for 4.49 dollars or packs of 24 for 9.99 dollars. What is the cheapest way to get at least 30 boxes?
  5. Two groups of 20 and 25 children share a 72 dollar bus fairly. How much does each group pay?
  6. A budget of 50 dollars must feed 20 people. What is the most you can spend per person?
  7. A party plan is 6.50 dollars over budget. You could drop balloons (saves 2.99), switch to a cheaper cake (saves 4.00), or drop dessert entirely (saves 7.20). Which single change fixes the budget? Which pair of changes would also work?
  8. A club has 45 dollars and wants to buy a game for 39.99 dollars. Snacks cost 1.25 dollars per member and there are 8 members. Can they afford both? If not, how much more do they need?
Answers
  1. Spent: \(12.50 + 9.75 + 8.20 = 30.45\) dollars. Left: \(40.00 - 30.45 = 9.55\) dollars.
  2. Rounded up: \(7 + 4 + 10 + 3 = 24\) dollars, which is under 25, so yes. Exact: \(6.40 + 3.15 + 9.99 + 2.50 = 22.04\) dollars.
  3. \(30 \div 8 = 3.75\), so round up to 4 pizzas (32 slices). Cost: \(4 \times 12.99 = 51.96\) dollars.
  4. Three packs of 10: \(3 \times 4.49 = 13.47\) dollars. One 24-pack and one 10-pack: \(9.99 + 4.49 = 14.48\) dollars. Two 24-packs: \(2 \times 9.99 = 19.98\) dollars. Three 10-packs are cheapest, 13.47 dollars, even though the 24-pack has a lower price per box (\(9.99 \div 24 \approx 0.42\) against \(4.49 \div 10 \approx 0.45\)), because you would pay for many boxes you do not need.
  5. \(20 : 25 = 4 : 5\). Boxes: 9. One box: \(72 \div 9 = 8\) dollars. The groups pay 32 and 40 dollars. (Check: \(32 \div 20 = 1.60\) and \(40 \div 25 = 1.60\) dollars per child.)
  6. \(50 \div 20 = 2.50\) dollars per person.
  7. Only dropping dessert works alone: 7.20 is more than 6.50. Balloons (2.99) or the cheaper cake (4.00) alone are not enough. Balloons and the cheaper cake together save \(2.99 + 4.00 = 6.99\) dollars, which also works.
  8. Snacks: \(8 \times 1.25 = 10\) dollars. Total: \(39.99 + 10 = 49.99\) dollars. That is more than 45, so no. They need \(49.99 - 45 = 4.99\) dollars more.

Common mistake

Rounding down when you must have enough. If you need 24 juice boxes and they come in tens, 2.4 packs means buying 3 packs, not 2. And when you estimate costs, round up, so the estimate is safe.

Forgetting small costs. Plates, napkins, a candle, a bag of ice: each is small, but together they can push a tight budget over. List everything before you shop.

Where it leads