Practice · A birthday trick is just 100x+y in disguise · Sensim Math

Variant 1 easy

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $8$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $25$ , add $8$ , multiply by $4$ .
Then add the day $y$ and subtract $40$ ; the result is (A).
Adding $8$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 25, plus 8, times 4

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

4

.

(25x+8)\times4=100x+32

25\times4=100

and

8\times4=32

.

2Add the day, subtract 40

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+32+y-40=100x+y-8

Combine the constants:

32-40=-8

.

3Grace adds 8

#13 Convert to Algebra 7.EE.A.1

Adding

8

cancels the

-8

, giving (B).

100x+y-8+8=100x+y

The

+8

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For February

17

it is

217

: month

2

, day

17

.

100\times2+17=217

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-8

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with February $17$ : $(25\times2+8)\times4=58\times4=232$ , then $232+17-40=209$ , and $100\times2+17-8=209$ . Adding $8$ gives $217$ , which reads as $2/17$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+32+y-40$ , so she must add back $40-32=8$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $4$ over $25x+8$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 2 easy

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $35$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $20$ , add $3$ , multiply by $5$ .
Then add the day $y$ and subtract $50$ ; the result is (A).
Adding $35$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 20, plus 3, times 5

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

5

.

(20x+3)\times5=100x+15

20\times5=100

and

3\times5=15

.

2Add the day, subtract 50

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+15+y-50=100x+y-35

Combine the constants:

15-50=-35

.

3Grace adds 35

#13 Convert to Algebra 7.EE.A.1

Adding

35

cancels the

-35

, giving (B).

100x+y-35+35=100x+y

The

+35

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For December

3

it is

1203

: month

12

, day

03

.

100\times12+3=1203

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-35

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with December $3$ : $(20\times12+3)\times5=243\times5=1215$ , then $1215+3-50=1168$ , and $100\times12+3-35=1168$ . Adding $35$ gives $1203$ , which reads as $12/03$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+15+y-50$ , so she must add back $50-15=35$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $5$ over $20x+3$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 3 easy

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $2$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $50$ , add $4$ , multiply by $2$ .
Then add the day $y$ and subtract $10$ ; the result is (A).
Adding $2$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 50, plus 4, times 2

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

2

.

(50x+4)\times2=100x+8

50\times2=100

and

4\times2=8

.

2Add the day, subtract 10

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+8+y-10=100x+y-2

Combine the constants:

8-10=-2

.

3Grace adds 2

#13 Convert to Algebra 7.EE.A.1

Adding

2

cancels the

-2

, giving (B).

100x+y-2+2=100x+y

The

+2

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For April

19

it is

419

: month

4

, day

19

.

100\times4+19=419

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-2

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with April $19$ : $(50\times4+4)\times2=204\times2=408$ , then $408+19-10=417$ , and $100\times4+19-2=417$ . Adding $2$ gives $419$ , which reads as $4/19$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+8+y-10$ , so she must add back $10-8=2$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $2$ over $50x+4$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 4 easy

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $40$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $10$ , add $6$ , multiply by $10$ .
Then add the day $y$ and subtract $100$ ; the result is (A).
Adding $40$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 10, plus 6, times 10

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

10

.

(10x+6)\times10=100x+60

10\times10=100

and

6\times10=60

.

2Add the day, subtract 100

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+60+y-100=100x+y-40

Combine the constants:

60-100=-40

.

3Grace adds 40

#13 Convert to Algebra 7.EE.A.1

Adding

40

cancels the

-40

, giving (B).

100x+y-40+40=100x+y

The

+40

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For July

25

it is

725

: month

7

, day

25

.

100\times7+25=725

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-40

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with July $25$ : $(10\times7+6)\times10=76\times10=760$ , then $760+25-100=685$ , and $100\times7+25-40=685$ . Adding $40$ gives $725$ , which reads as $7/25$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+60+y-100$ , so she must add back $100-60=40$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $10$ over $10x+6$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 5 medium

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $60$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $5$ , add $7$ , multiply by $20$ .
Then add the day $y$ and subtract $200$ ; the result is (A).
Adding $60$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 5, plus 7, times 20

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

20

.

(5x+7)\times20=100x+140

5\times20=100

and

7\times20=140

.

2Add the day, subtract 200

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+140+y-200=100x+y-60

Combine the constants:

140-200=-60

.

3Grace adds 60

#13 Convert to Algebra 7.EE.A.1

Adding

60

cancels the

-60

, giving (B).

100x+y-60+60=100x+y

The

+60

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For October

4

it is

1004

: month

10

, day

04

.

100\times10+4=1004

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-60

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with October $4$ : $(5\times10+7)\times20=57\times20=1140$ , then $1140+4-200=944$ , and $100\times10+4-60=944$ . Adding $60$ gives $1004$ , which reads as $10/04$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+140+y-200$ , so she must add back $200-140=60$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $20$ over $5x+7$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 6 medium

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $70$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $10$ , add $13$ , multiply by $10$ .
Then add the day $y$ and subtract $200$ ; the result is (A).
Adding $70$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 10, plus 13, times 10

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

10

.

(10x+13)\times10=100x+130

10\times10=100

and

13\times10=130

.

2Add the day, subtract 200

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+130+y-200=100x+y-70

Combine the constants:

130-200=-70

.

3Grace adds 70

#13 Convert to Algebra 7.EE.A.1

Adding

70

cancels the

-70

, giving (B).

100x+y-70+70=100x+y

The

+70

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For June

11

it is

611

: month

6

, day

11

.

100\times6+11=611

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-70

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with June $11$ : $(10\times6+13)\times10=73\times10=730$ , then $730+11-200=541$ , and $100\times6+11-70=541$ . Adding $70$ gives $611$ , which reads as $6/11$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+130+y-200$ , so she must add back $200-130=70$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $10$ over $10x+13$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 7 medium

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $25$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $4$ , add $7$ , multiply by $25$ .
Then add the day $y$ and subtract $200$ ; the result is (A).
Adding $25$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 4, plus 7, times 25

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

25

.

(4x+7)\times25=100x+175

4\times25=100

and

7\times25=175

.

2Add the day, subtract 200

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+175+y-200=100x+y-25

Combine the constants:

175-200=-25

.

3Grace adds 25

#13 Convert to Algebra 7.EE.A.1

Adding

25

cancels the

-25

, giving (B).

100x+y-25+25=100x+y

The

+25

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For March

14

it is

314

: month

3

, day

14

.

100\times3+14=314

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-25

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with March $14$ : $(4\times3+7)\times25=19\times25=475$ , then $475+14-200=289$ , and $100\times3+14-25=289$ . Adding $25$ gives $314$ , which reads as $3/14$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+175+y-200$ , so she must add back $200-175=25$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $25$ over $4x+7$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 8 medium

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $70$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $5$ , add $9$ , multiply by $20$ .
Then add the day $y$ and subtract $250$ ; the result is (A).
Adding $70$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 5, plus 9, times 20

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

20

.

(5x+9)\times20=100x+180

5\times20=100

and

9\times20=180

.

2Add the day, subtract 250

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+180+y-250=100x+y-70

Combine the constants:

180-250=-70

.

3Grace adds 70

#13 Convert to Algebra 7.EE.A.1

Adding

70

cancels the

-70

, giving (B).

100x+y-70+70=100x+y

The

+70

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For August

21

it is

821

: month

8

, day

21

.

100\times8+21=821

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-70

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with August $21$ : $(5\times8+9)\times20=49\times20=980$ , then $980+21-250=751$ , and $100\times8+21-70=751$ . Adding $70$ gives $821$ , which reads as $8/21$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+180+y-250$ , so she must add back $250-180=70$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $20$ over $5x+9$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 9 hard

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $60$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $5$ , add $12$ , multiply by $20$ .
Then add the day $y$ and subtract $300$ ; the result is (A).
Adding $60$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 5, plus 12, times 20

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

20

.

(5x+12)\times20=100x+240

5\times20=100

and

12\times20=240

.

2Add the day, subtract 300

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+240+y-300=100x+y-60

Combine the constants:

240-300=-60

.

3Grace adds 60

#13 Convert to Algebra 7.EE.A.1

Adding

60

cancels the

-60

, giving (B).

100x+y-60+60=100x+y

The

+60

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For November

6

it is

1106

: month

11

, day

06

.

100\times11+6=1106

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-60

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with November $6$ : $(5\times11+12)\times20=67\times20=1340$ , then $1340+6-300=1046$ , and $100\times11+6-60=1046$ . Adding $60$ gives $1106$ , which reads as $11/06$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+240+y-300$ , so she must add back $300-240=60$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $20$ over $5x+12$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 10 hard

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $50$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $5$ , add $20$ , multiply by $20$ .
Then add the day $y$ and subtract $450$ ; the result is (A).
Adding $50$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 5, plus 20, times 20

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

20

.

(5x+20)\times20=100x+400

5\times20=100

and

20\times20=400

.

2Add the day, subtract 450

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+400+y-450=100x+y-50

Combine the constants:

400-450=-50

.

3Grace adds 50

#13 Convert to Algebra 7.EE.A.1

Adding

50

cancels the

-50

, giving (B).

100x+y-50+50=100x+y

The

+50

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For January

28

it is

128

: month

1

, day

28

.

100\times1+28=128

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-50

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with January $28$ : $(5\times1+20)\times20=25\times20=500$ , then $500+28-450=78$ , and $100\times1+28-50=78$ . Adding $50$ gives $128$ , which reads as $1/28$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+400+y-450$ , so she must add back $450-400=50$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $20$ over $5x+20$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 11 hard

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $125$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $4$ , add $15$ , multiply by $25$ .
Then add the day $y$ and subtract $500$ ; the result is (A).
Adding $125$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 4, plus 15, times 25

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

25

.

(4x+15)\times25=100x+375

4\times25=100

and

15\times25=375

.

2Add the day, subtract 500

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+375+y-500=100x+y-125

Combine the constants:

375-500=-125

.

3Grace adds 125

#13 Convert to Algebra 7.EE.A.1

Adding

125

cancels the

-125

, giving (B).

100x+y-125+125=100x+y

The

+125

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For September

30

it is

930

: month

9

, day

30

.

100\times9+30=930

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-125

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with September $30$ : $(4\times9+15)\times25=51\times25=1275$ , then $1275+30-500=805$ , and $100\times9+30-125=805$ . Adding $125$ gives $930$ , which reads as $9/30$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+375+y-500$ , so she must add back $500-375=125$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $25$ over $4x+15$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

Variant 12 hard

Read the conversation below and answer the question.

Let the birth month be $x$ and the birth day be $y$ . Write (A) and (B) as simplified expressions in $x$ and $y$ .

Your answer (check it yourself)

Show solution

1 · Understandwhat's really being asked

Grace has Hannah run her birth month and day through a chain of operations, then secretly adds $100$ . Write Hannah's result (A) and Grace's adjusted number (B) as expressions in the month $x$ and day $y$ .

Givens

Month $x$ : multiply by $2$ , add $30$ , multiply by $50$ .
Then add the day $y$ and subtract $1600$ ; the result is (A).
Adding $100$ to (A) gives (B).

Unknowns

(A) and (B) as simplified expressions in $x$ and $y$ .

Constraints

$x$ is a month number from $1$ to $12$ and $y$ a day number from $1$ to $31$ .

2 · Planchoose the strategy

#13 Convert to Algebra · also uses: #7 Identify Subproblems#5 Look for a Pattern

Translate each spoken step into algebra, in order, and simplify with the distributive property. The simplified form shows why Grace can read off the birthday.

3 · Execute4 carry out the plan

1Times 2, plus 30, times 50

#13 Convert to Algebra 7.EE.A.1

Write the month steps and distribute the

50

.

(2x+30)\times50=100x+1500

2\times50=100

and

30\times50=1500

.

2Add the day, subtract 1600

#7 Identify Subproblems 7.EE.A.1

This is Hannah's answer (A).

100x+1500+y-1600=100x+y-100

Combine the constants:

1500-1600=-100

.

3Grace adds 100

#13 Convert to Algebra 7.EE.A.1

Adding

100

cancels the

-100

, giving (B).

100x+y-100+100=100x+y

The

+100

is exactly what undoes the leftover constant.

4Why it reveals the birthday

#5 Look for a Pattern 7.EE.A.2

100x+y

puts the month in the hundreds (and thousands) place and the day in the last two digits. For May

9

it is

509

: month

5

, day

09

.

100\times5+9=509

Rewriting the steps as

100x+y

exposes the trick.

Answer: (A)

100x+y-100

, (B)

100x+y

4 · Reviewdoes it hold up?

Check with May $9$ : $(2\times5+30)\times50=40\times50=2000$ , then $2000+9-1600=409$ , and $100\times5+9-100=409$ . Adding $100$ gives $509$ , which reads as $5/09$ .

Another way: Work backwards from what Grace needs: to read the date she wants $100x+y$ . The steps produce $100x+1500+y-1600$ , so she must add back $1600-1500=100$ .

Standardsmin grade 7

7.EE.A.1 Apply properties of operations to add, subtract, factor, and expand linear expressions — Distributing $50$ over $2x+30$ and combining constants.
7.EE.A.2 Rewrite an expression in different forms to shed light on the problem — Seeing that $100x+y$ encodes the month and the day as digits.

💡Takeaway. A magic trick with numbers is often just algebra in disguise; simplify it and the secret shows.

A birthday trick is just 100x+y in disguise · 12 practice problems

Generated variants — 12

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