Story problems with dollars and cents

When you reach for it

A child who can count a handful of coins knows how much money is in the hand. Money stories ask the next question. How much is there after more comes in? How much is left after something is bought? How much more is needed before something can be bought? Which of two things costs more, and by how much? Each of these is an adding or subtracting story, the same kinds a child has met with counters and cubes, only now the things being counted are cents or dollars.

Two things are new.

• The answer has a unit, and the unit has a sign. An answer of 18 is not finished. It is either 18\text{¢} or \text{\textdollar}18, and those are very different amounts. The story says which.
• The answer can be checked against the world. Money is something a child already has a feel for. A pack of stickers does not cost \text{\textdollar}80, and nobody has more left after spending than before. That feel catches a wrong operation quickly.

The lesson runs over three days.

• Day 1: one-step stories in all and left. Adding amounts that come in and taking away amounts spent, in cents or in dollars, with the right sign on the answer.
• Day 2: how much more. How much more is needed to buy something, how much more one amount is than another, and how much there was at the start.
• Day 3: two-step stories. Buying two things and getting change, or earning twice and then checking against a goal.

The child already counts coins into cents with Counting mixed coins, and knows what each coin and bill is worth, how the two signs are written, and how change is made by counting up, from Money. Amounts here stay within 100 cents, or are whole dollars within 100 dollars. The adding and subtracting itself, tens first and then ones, is the work of Regrouping in addition and Regrouping in subtraction.

The procedure

Part 1: One unit, one sign

Cents with cents, dollars with dollars. Before any adding, ask what is being counted. In a story about a bottle return machine, the amounts are cents. In a story about buying a fish tank, they are dollars. Every number in one number sentence must count the same unit. Then the sum or difference counts that unit too, so the answer takes the same sign.

This is the same rule as adding tens to tens and ones to ones. 3 tens and 4 ones are not 7 of anything; 3 dollars and 4 cents are not 7 of anything either.

Coins become cents first. A story may give coins rather than an amount: “the machine gave her 3 dimes and 1 nickel”. Count the coins into one amount, 35\text{¢}, before it goes into the number sentence. Bills become dollars the same way.

Writing the sign. The cent sign goes after the number and the dollar sign goes before it.

An amount gets one sign, never two. There is no amount written \text{\textdollar}84\text{¢}: it would have to mean dollars and cents at once. In the number sentence itself, the numbers may be written bare, 46 + 38 = 84, as long as the child knows what they count. The sign goes back on in the answer sentence: “She has 84\text{¢}.”

One dollar in a cents story. When a story mixes a \text{\textdollar}1 bill with prices in cents, change the dollar to 100\text{¢} so every number is in cents. That is the fact from Money that one dollar is worth exactly 100 cents.

In all and left. When money comes in and joins what is there, add. When money is spent or given away, the amount spent is taken from what there was, so subtract. The words of the story are a guide, not a rule: “more” can appear in an adding story (“she got 30\text{¢} more”) and in a subtracting one (“how much more does she need?”). What decides it is what happens to the money.

Checking against the story. Two checks fit every money story.

• Does the size make sense? After spending, what is left must be less than what there was. After getting more, the total must be more than either part.
• Does it undo? For a subtraction, add the answer back to what was taken and see the start. For an addition, take one part away from the total and see the other part.

Part 2: How much more

How much more is needed. Someone has some money and wants something that costs more. The money in hand and the money still needed together make the price:

The missing part can be found two ways. Count up from what is in hand to the price, the way change is counted in Money: that finds \square directly. Or subtract: \text{price} - \text{have}. Both give the same number, because the two sentences say the same thing about the same three amounts.

How much more one amount is than another. Comparing two prices, or two people’s savings, asks for the difference. Line the two amounts up from the same start; the smaller one covers part of the bigger one, and the uncovered part is how much more. A number line or two bars drawn side by side, as in Bar models for story problems, show it at a glance. The answer is \text{bigger} - \text{smaller}.

When the start is missing. “She had some money, got 35\text{¢}, and now has 92\text{¢}.” The start is the unknown:

The total and one part are known, so the other part is the total take away the known part. Here the start is not found by adding the two numbers given: the total already includes the 35\text{¢}. (When money was spent instead, the start is the total: what was spent and what is left together make it, so then the two numbers are added.)

Part 3: Two steps

Many shopping stories hide a question inside the one that is asked. “She buys two things and pays with a dollar. What is her change?” Before the change can be found, the cost of the two things together must be found. Find the hidden amount first, write it down with its sign, then use it in the second step. The child should say in words what the first number means (“73\text{¢} is what the two things cost”) before going on, so the second step uses it for the right job.

The unit stays the same through both steps. If the first step is in cents, the second is in cents.

A check still closes the work, and it can go back through both steps: add the change to the cost, then split the cost back into its two prices.

Worked examples

Examples 1 and 2 go with Day 1, Examples 3 to 5 with Day 2, and Examples 6 and 7 with Day 3.

Boundaries

• One unit per number sentence. Cents are added to cents and dollars to dollars. A story that pays a \text{\textdollar}1 bill toward a price in cents is turned all into cents first, by writing the dollar as 100\text{¢} (Example 6).
• One sign per amount. An amount here is written with the cent sign after it or the dollar sign before it, never both. An amount of dollars and cents together, such as 3 dollars and 40 cents, is said in words for now. Writing it as one number needs the decimal point of Money written as decimals.
• Amounts stay within 100 cents or 100 dollars. A story whose total runs past 100\text{¢} makes a dollar and some cents, which is the case above.
• Spending needs enough money. In a “how much is left” story, the price must be no more than what there is. If it is more, nothing is bought, and the question becomes how much more is needed (Example 3).
• “How much more” can mean two things. It can ask what is still needed to reach a price, or how much bigger one amount is than another. Both are answered by the missing part, the bigger amount take away the smaller, but the answer sentence must say which: “she needs 17\text{¢} more” or “the large tank costs \text{\textdollar}35 more”.
• The words of a story do not choose the operation. “More” appears in adding stories and in subtracting ones. What chooses is which amount is the whole and which are its parts: parts that join make the whole, so add; the whole take away a known part leaves the other part, so subtract.

Where this goes next

The same stories come back with larger numbers once the child adds and subtracts within 1{,}000, and in Two-step addition and subtraction problems, with every kind of quantity rather than only money. Counting up to find what is still needed is the same move as Counting up to find a difference, which carries it across hundreds. Writing those amounts as one number, such as \text{\textdollar}3.40, with the cents as hundredths of a dollar, is Money written as decimals; the stories then mix dollars and cents freely, because the decimal point keeps the units lined up the way the cent sign and the dollar sign do here. Later still, comparing what the same thing costs in two sizes is Unit price and comparing rates.

Exercises

In all and left (Day 1)

1. At the bottle return, Mahealani gets 25\text{¢} for a bag of cans and 40\text{¢} for a bag of bottles. How much does she get in all?
2. Zhanibek has \text{\textdollar}43. At the aquarium shop he buys a tank heater for \text{\textdollar}21. How much money does he have left?
3. Dobrinka has 80\text{¢}. She buys a plastic plant for her fish bowl for 46\text{¢}. How much money does she have left?
4. At the school carnival, the face painting table takes in \text{\textdollar}28 in the morning and \text{\textdollar}34 in the afternoon. How much does it take in altogether?
5. Mahealani has 2 quarters. The bottle return machine gives her 3 dimes and 1 nickel. How much money does she have now?
6. Write each amount with the right sign: twelve dollars; sixty cents; ninety-nine cents; forty dollars.
7. At the school carnival, a sticker costs 30 and a goldfish tank costs 30. One price is in cents and the other in dollars. Which is which? Write each with its sign.
8. Zhanibek had 70\text{¢}. He spent 38\text{¢} on a bag of popcorn at the carnival. How much does he have left? Check your answer by adding.

How much more (Day 2)

9. At the aquarium shop a small fish net costs 60\text{¢}. Dobrinka has 45\text{¢}. How much more does she need?
10. A tank light costs \text{\textdollar}55. Zhanibek has \text{\textdollar}38. How much more does he need?
11. At the bottle return, Mahealani gets 64\text{¢} and Dobrinka gets 39\text{¢}. How much more does Mahealani get?
12. At the school carnival, the ring toss takes in \text{\textdollar}52 and the bean bag throw takes in \text{\textdollar}29. How much more does the ring toss take in?
13. Zhanibek had some money. His aunt gave him \text{\textdollar}15, and now he has \text{\textdollar}41. How much did he have before?
14. Dobrinka had 28\text{¢}. After a trip to the bottle return she has 75\text{¢}. How much did the machine give her?
15. Mahealani had some money. She spent 35\text{¢} on a snow cone and has 40\text{¢} left. How much did she have to start with?
16. A bag of gravel costs 90\text{¢}. Zhanibek has 2 quarters and 2 dimes. How much more does he need?

Two steps (Day 3)

17. Dobrinka buys a fish magnet for 25\text{¢} and a pencil for 32\text{¢}. She pays with a \text{\textdollar}1 bill. How much change does she get?
18. Zhanibek has \text{\textdollar}70. At the aquarium shop he buys a filter for \text{\textdollar}18 and a thermometer for \text{\textdollar}25. How much does he have left?
19. Mahealani gets 35\text{¢} at the bottle return on Monday and 40\text{¢} on Tuesday. She spends 50\text{¢} on a juice box. How much does she have left?
20. At the school carnival, the lemonade stand takes in \text{\textdollar}45. It spends \text{\textdollar}12 on more lemons and cups, then takes in \text{\textdollar}26 more. How much money does it have now?
21. Mahealani has 1 quarter and 2 dimes. Dobrinka has 3 dimes and 1 nickel. Together, do they have enough for a 75\text{¢} bag of fish food? How much is over or short?
22. Zhanibek buys a book for \text{\textdollar}7 and a puzzle for \text{\textdollar}9. He pays with a \text{\textdollar}20 bill. How much change does he get?

Explain

23. A box of crayons costs forty-five cents. Why is it wrong to write the price as \text{\textdollar}45?
24. Dobrinka started with 90\text{¢} and bought something. Could she have 120\text{¢} left? Without doing any adding or subtracting, how can you tell?
25. In a “how much more is needed” story, why does counting up from what you have give the same answer as subtracting what you have from the price?