Build the decimal closest to a target from cards

5.NBT.B.74.NF.C.7 · take · grade 5

Archetype: Get Closest to a Target Value · step in a 4-type progression

Using all four number cards exactly once each, you want to form a number with two decimal places (a hundredths decimal). Among all such numbers you can make, find the one closest to $30$ .

Number cards: $3$ , $2$ , $4$ , $7$

Show solution

Understand

Use each of the cards 3, 2, 4, 7 exactly once to build a number that has two digits before the decimal point and two digits after it (a hundredths decimal). Among all such numbers, find the one whose value is nearest to 30.

Givens

The four cards are 3, 2, 4, and 7.
Each card is used exactly once.
The number has the form (two whole-number digits).(two decimal digits), e.g. 27.43.
The target value is 30.

Unknowns

The number, built from all four cards, that is closest to 30.

Constraints

All four cards must be used, none repeated.
Exactly two digits sit before the decimal point and two after.

Plan

#6 Guess and Check · also uses: #2 Make a Systematic List

Being closest to 30 means the whole-number part must be just below 30 (a 27.xx number) or just above 30 (a 32.xx number), since no card pair makes 30 itself. So we only need to check those few candidates and compare how far each is from 30.

Execute

#2 Make a Systematic List 5.NBT.A.1

To land near 30, the tens digit should be 2 or 3. With a 2 in the tens place the largest whole part is 27; with a 3 in the tens place the smallest whole part is 32. These bracket 30 from below and above, so only 27.xx and 32.xx numbers can be closest.

27.\square\square \le 30 \le 32.\square\square

Reading the tens place tells you the rough size, just like ordering whole numbers.

#6 Guess and Check 4.NF.C.7

Using 2 and 7 as the whole part leaves 3 and 4 for the decimals: the closest is 27.43 (put the larger remaining value high). Using 3 and 2 as the whole part leaves 4 and 7 for the decimals: the closest to 30 is 32.47 (make the decimal part smallest).

27.43 \quad\text{and}\quad 32.47

Pick the digit arrangement that nudges each candidate toward 30.

#6 Guess and Check 5.NBT.B.7

Find how far each candidate is from 30 by subtracting. 30 - 27.43 = 2.57, while 32.47 - 30 = 2.47. Since 2.47 is smaller than 2.57, 32.47 is nearer to 30.

30-27.43=2.57,\ \ 32.47-30=2.47,\ \ 2.47<2.57

The smaller the difference, the closer the number - so compute both gaps and pick the smaller.

Answer: 32.47

Review

Both numbers use the cards 3, 2, 4, 7 once each. 32.47 sits 2.47 above 30 and 27.43 sits 2.57 below 30, so 32.47 wins by 0.10. No 27.xx or 32.xx arrangement is closer (27.43 is the largest 27.xx, 32.47 is the smallest 32.xx).

Make a Systematic List: list every closest-of-its-block candidate (24.73, 27.43, 32.47, 34.27, 42.37, ...) and read off the one with the smallest gap to 30, which is again 32.47.

Standards · min grade 5

5.NBT.B.7 Add, subtract, multiply, and divide decimals to hundredths — Subtracting to find each candidate's distance from 30 (2.57 and 2.47).
5.NBT.A.1 Recognize that a digit in one place represents ten times as much as to its right — Choosing the tens digit (2 or 3) to make the whole part near 30.
4.NF.C.7 Compare two decimals to hundredths by reasoning about their size — Arranging the decimal digits to make each candidate as close to 30 as possible.

💡 Closest to 30 means smallest gap - so just check the numbers right below and right above 30 and subtract!